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83 matrix notation

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A matrix is a rectangular table of numbers that can represent a system of linear equations. Matrix notation uses row operations like switching rows, multiplying rows by constants, and adding multiples of rows to other rows to transform the matrix into upper triangular form, from which the solution to the system of equations can be determined. The three main row operations are: 1) switching rows, 2) multiplying a row by a constant, and 3) adding a multiple of one row to another row.

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Matrix Notation A matrix is a rectangular table of numbers.
Matrix Notation A matrix is a rectangular table of numbers.
Matrix Notation A matrix is a rectangular table of numbers. For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4 are matrices.
Matrix Notation A matrix is a rectangular table of numbers. For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4 are matrices. Systems of linear equations can be put into matrices then solved using matrix notation and elimination method.
Matrix Notation A matrix is a rectangular table of numbers. For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4 are matrices. Systems of linear equations can be put into matrices then solved using matrix notation and elimination method. For example, if we put the following system into a matrix, x + 4y = –7 Eq. 1 {2x – 3y = 8 Eq. 2
Matrix Notation A matrix is a rectangular table of numbers. For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4 are matrices. Systems of linear equations can be put into matrices then solved using matrix notation and elimination method. For example, if we put the following system into a matrix, x + 4y = –7 Eq. 1 {2x – 3y = 8 Eq. 2 we get 1 4 -7 2 -3 8
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers.
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations.
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 2 -3 8
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7 II. Multiply Row i by a nonzero constant k, notated as k*Ri.
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7 II. Multiply Row i by a nonzero constant k, notated as k*Ri. For example: 1 4 -7 2 -3 8
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7 II. Multiply Row i by a nonzero constant k, notated as k*Ri. For example: 1 4 -7 -3*R2 2 -3 8
Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7 II. Multiply Row i by a nonzero constant k, notated as k*Ri. For example: 1 4 -7 -3*R2 1 4 -7 2 -3 8 -6 9 -24
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, 1 4 -7 2 -3 8
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, 1 4 -7 -2*R1 add  R2 2 -3 8
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 2 -3 8
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system.
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation 1. Apply row operations to transform the matrix to the upper diagonal form where all the entries below the main diagonal (the lower left triangular region) are 0.
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation 1. Apply row operations to transform the matrix to the upper diagonal form where all the entries below the main diagonal (the lower left triangular region) are 0. * * * * * * * * * * * *
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation 1. Apply row operations to transform the matrix to the upper diagonal form where all the entries below the main diagonal (the lower left triangular region) are 0. * * * * Row operations * * * * * * * *
Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation 1. Apply row operations to transform the matrix to the upper diagonal form where all the entries below the main diagonal (the lower left triangular region) are 0. * * * * Row operations * * * * * * * * 0 0 * * * 0 * * * * * *
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable.
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable.
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions.
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix:
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: 1 4 -7 2 -3 8
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: 1 4 -7 2 -3 8
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: 1 4 -7 -2*R1 add  R2 2 -3 8
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 2 -3 8
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2 Go up one row to R1 and set y = -2:
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2 Go up one row to R1 and set y = -2: x + 4y = -7
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2 Go up one row to R1 and set y = -2: x + 4y = -7 x + 4(-2) = -7
Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2 Go up one row to R1 and set y = -2: x + 4y = -7 x + 4(-2) = -7 x=1 Solution : (1, -2)
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix:
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: 2 3 3 13 1 2 2 8 3 2 3 13
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: 2 3 3 13 R1 R2 1 2 2 8 3 2 3 13
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -2R1 add R2
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -2R1 add R2
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -2R1 add R2 1 2 2 8 0 -1 -1 -3 3 2 3 13
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -2R1 add R2 1 2 2 8 0 -1 -1 -3 -3* R1 add R3 3 2 3 13
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 0 -1 -1 -3 -3* R1 add R3 3 2 3 13
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 2 2 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 2 2 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11 -4*R2 add R3
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 0 24 2 4 12 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11 -4*R2 add R3
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 0 24 2 4 12 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11 1 2 2 8 -4*R2 add R3 0 -1 -1 -3 0 0 1 1
Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 0 2 4 2 12 4 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11 1 2 2 8 -4*R2 add R3 0 -1 -1 -3 0 0 1 1 It's in upper diagonal form. Ready to solve.
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1.
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1. From R2, -y – z = -3
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1. From R2, -y – z = -3 -y – (1) = -3
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1. From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1, From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y From R1, we get x + 2y + 2z = 8
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1, From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y From R1, we get x + 2y + 2z = 8 x + 2(2) + 2(1) = 8
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1, From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y From R1, we get x + 2y + 2z = 8 x + 2(2) + 2(1) = 8 x+6=8
Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1, From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y From R1, we get x + 2y + 2z = 8 x + 2(2) + 2(1) = 8 x+6=8 x=2 So the solution is (2, 2, 1)

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83 matrix notation

  • 1. Matrix Notation A matrix is a rectangular table of numbers.
  • 2. Matrix Notation A matrix is a rectangular table of numbers.
  • 3. Matrix Notation A matrix is a rectangular table of numbers. For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4 are matrices.
  • 4. Matrix Notation A matrix is a rectangular table of numbers. For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4 are matrices. Systems of linear equations can be put into matrices then solved using matrix notation and elimination method.
  • 5. Matrix Notation A matrix is a rectangular table of numbers. For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4 are matrices. Systems of linear equations can be put into matrices then solved using matrix notation and elimination method. For example, if we put the following system into a matrix, x + 4y = –7 Eq. 1 {2x – 3y = 8 Eq. 2
  • 6. Matrix Notation A matrix is a rectangular table of numbers. For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4 are matrices. Systems of linear equations can be put into matrices then solved using matrix notation and elimination method. For example, if we put the following system into a matrix, x + 4y = –7 Eq. 1 {2x – 3y = 8 Eq. 2 we get 1 4 -7 2 -3 8
  • 7. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers.
  • 8. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations.
  • 9. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices
  • 10. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj
  • 11. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 2 -3 8
  • 12. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7
  • 13. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7 II. Multiply Row i by a nonzero constant k, notated as k*Ri.
  • 14. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7 II. Multiply Row i by a nonzero constant k, notated as k*Ri. For example: 1 4 -7 2 -3 8
  • 15. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7 II. Multiply Row i by a nonzero constant k, notated as k*Ri. For example: 1 4 -7 -3*R2 2 -3 8
  • 16. Matrix Notation Each row of the matrix corresponds to an equation and each column corresponds to a variable except that the last column corresponds to numbers. Operations of the equations correspond to operations of rows in the matrices. There’re three main operations. The Three Row Operations for Matrices I. Switching rows, notated as Row i Row j, or Ri Rj For example, 1 4 -7 R1 R2 2 -3 8 2 -3 8 1 4 -7 II. Multiply Row i by a nonzero constant k, notated as k*Ri. For example: 1 4 -7 -3*R2 1 4 -7 2 -3 8 -6 9 -24
  • 17. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.
  • 18. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, 1 4 -7 2 -3 8
  • 19. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, 1 4 -7 -2*R1 add  R2 2 -3 8
  • 20. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 2 -3 8
  • 21. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22
  • 22. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system.
  • 23. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation
  • 24. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation 1. Apply row operations to transform the matrix to the upper diagonal form where all the entries below the main diagonal (the lower left triangular region) are 0.
  • 25. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation 1. Apply row operations to transform the matrix to the upper diagonal form where all the entries below the main diagonal (the lower left triangular region) are 0. * * * * * * * * * * * *
  • 26. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation 1. Apply row operations to transform the matrix to the upper diagonal form where all the entries below the main diagonal (the lower left triangular region) are 0. * * * * Row operations * * * * * * * *
  • 27. Matrix Notation III. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”. For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 Fact: Performing row operations on a matrix does not change the solution of the system. Elimination Method in Matrix Notation 1. Apply row operations to transform the matrix to the upper diagonal form where all the entries below the main diagonal (the lower left triangular region) are 0. * * * * Row operations * * * * * * * * 0 0 * * * 0 * * * * * *
  • 28. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable.
  • 29. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable.
  • 30. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions.
  • 31. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2
  • 32. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix:
  • 33. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: 1 4 -7 2 -3 8
  • 34. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: 1 4 -7 2 -3 8
  • 35. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: 1 4 -7 -2*R1 add  R2 2 -3 8
  • 36. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 2 -3 8
  • 37. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22
  • 38. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2
  • 39. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2 Go up one row to R1 and set y = -2:
  • 40. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2 Go up one row to R1 and set y = -2: x + 4y = -7
  • 41. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2 Go up one row to R1 and set y = -2: x + 4y = -7 x + 4(-2) = -7
  • 42. Matrix Notation 2. Starting from the bottom row, get the answer for one of the variable. Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working up the rows to extract all solutions. Example A. Solve using matrix notation. x + 4y = -7 Eq. 1 { 2x – 3y = 8 Eq. 2 Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22 From the bottom R2: -11y = 22  y = -2 Go up one row to R1 and set y = -2: x + 4y = -7 x + 4(-2) = -7 x=1 Solution : (1, -2)
  • 43. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3
  • 44. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix:
  • 45. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: 2 3 3 13 1 2 2 8 3 2 3 13
  • 46. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: 2 3 3 13 R1 R2 1 2 2 8 3 2 3 13
  • 47. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13
  • 48. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -2R1 add R2
  • 49. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -2R1 add R2
  • 50. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -2R1 add R2 1 2 2 8 0 -1 -1 -3 3 2 3 13
  • 51. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -2R1 add R2 1 2 2 8 0 -1 -1 -3 -3* R1 add R3 3 2 3 13
  • 52. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 0 -1 -1 -3 -3* R1 add R3 3 2 3 13
  • 53. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 2 2 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11
  • 54. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 2 2 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11 -4*R2 add R3
  • 55. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 0 24 2 4 12 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11 -4*R2 add R3
  • 56. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 0 24 2 4 12 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11 1 2 2 8 -4*R2 add R3 0 -1 -1 -3 0 0 1 1
  • 57. Matrix Notation Example B. Solve using matrix notation. E1 { 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3 Put the system into a matrix: -2 -4 -4 -16 2 3 3 13 R1 R2 1 2 2 8 1 2 2 8 2 3 3 13 3 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R2 1 2 2 8 1 0 2 4 2 12 4 8 0 -1 -1 -3 -3* R1 add R3 0 -1 -1 -3 3 2 3 13 0 -4 -3 -11 1 2 2 8 -4*R2 add R3 0 -1 -1 -3 0 0 1 1 It's in upper diagonal form. Ready to solve.
  • 58. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1
  • 59. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1.
  • 60. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1. From R2, -y – z = -3
  • 61. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1. From R2, -y – z = -3 -y – (1) = -3
  • 62. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1. From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y
  • 63. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1, From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y From R1, we get x + 2y + 2z = 8
  • 64. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1, From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y From R1, we get x + 2y + 2z = 8 x + 2(2) + 2(1) = 8
  • 65. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1, From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y From R1, we get x + 2y + 2z = 8 x + 2(2) + 2(1) = 8 x+6=8
  • 66. Matrix Notation We have reduced the matrix. 2 3 3 13 1 2 2 8 1 2 2 8 0 -1 -1 -3 3 2 3 13 0 0 1 1 From R3, we get z = 1, From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y From R1, we get x + 2y + 2z = 8 x + 2(2) + 2(1) = 8 x+6=8 x=2 So the solution is (2, 2, 1)