Matrix2 english
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The document discusses matrices and their properties. It defines minors as the determinants of submatrices formed by deleting a row and column from the original matrix. Cofactors are defined as signed minors, with the sign determined by the parity of the sum of the row and column indices. The adjoint and inverse of matrices are discussed, with the inverse existing only for nonsingular matrices where the determinant is non-zero. Properties of the inverse include that the inverse of the inverse is the original matrix and that the inverse of a product is the product of the inverses in reverse order. Examples are provided to demonstrate calculating the adjoint, inverse and checking the inverse property.
Matrix2 english
- 1. MATRICES BY ALFIA MAGFIRONA D100102004 CIVIL ENGINEERING DEPARTEMENT ENGINEERING FACULTY MUHAMMADIYAH UNIVERSITY OF SURAKARTA
- 2. MATRICES - OPERATIONS MINORS If A is an n x n matrix and one row and one column are deleted, the resulting matrix is an (n-1) x (n-1) submatrix of A. The determinant of such a submatrix is called a minor of A and is designated by mij , where i and j correspond to the deleted row and column, respectively. mij is the minor of the element aij in A.
- 3. eg. a11 a12 a13 A a21 a22 a23 a31 a32 a33 Each element in A has a minor Delete first row and column from A . The determinant of the remaining 2 x 2 submatrix is the minor of a11 a22 a23 m11 a32 a33
- 4. Therefore the minor of a12 is: a21 a23 m12 a31 a33 And the minor for a13 is: a21 a22 m13 a31 a32
- 5. E. COFACTOR OF MATRIX If A is a square matrix, then the minor of its entry aij, also known as the i,j, or (i,j), or (i,j)th minor of A, is denoted by Mij and is defined to be the determinant of the submatrix obtained by removing from A its i-th row and j-th column. It follows: Cij ( 1)i j mij When the sum of a row number i and column j is even, cij = mij and when i+j is odd, cij =-mij c11 (i 1, j 1) ( 1)1 1 m11 m11 c12 (i 1, j 2) ( 1)1 2 m12 m12 1 3 c13 (i 1, j 3) ( 1) m13 m13
- 6. The Formula : C11 C12 C13 M 11 M 12 M 13 C21 C22 C23 M 21 M 22 M 23 C31 C32 C33 M 31 M 32 M 33
- 7. DETERMINANTS CONTINUED The determinant of an n x n matrix A can now be defined as A det A a11c11 a12c12 a1nc1n The determinant of A is therefore the sum of the products of the elements of the first row of A and their corresponding cofactors. (It is possible to define |A| in terms of any other row or column but for simplicity, the first row only is used)
- 8. Therefore the 2 x 2 matrix : a11 a12 A a21 a22 Has cofactors : c11 m11 a22 a22 And: c12 m12 a21 a21
- 9. For a 3 x 3 matrix: a11 a12 a13 A a21 a22 a23 a31 a32 a33 The cofactors of the first row are: a22 a23 c11 a22 a33 a23a32 a32 a33 a21 a23 c12 (a21a33 a23a31 ) a31 a33 a21 a22 c13 a21a32 a22 a31 a31 a32
- 10. F. ADJOINT OF MATRIX The adjoint matrix for 2 x 2 square matrix A= , so Adjoint of matrix A is Elements in the first diagonal of matrix is exchanged, and the second diagonal of matrix is just changed mark.
- 11. A= second diagonal of matrix first diagonal of matrix Adj A =
- 12. PROBLEM Find Adjoint of matrix We can use the formula of The adjoint matrix for 2 x 2 square matrix. So, Adj
- 13. The adjoint matrix for 3 x 3 square matrix OR
- 14. To determine the adjoint matrix for 3 x 3 square matrix is used cofactor matrix in each elements in the square of matrix.
- 15. It uses cofactor of matrix A1.1 to fill in fisrt rows of A and for the others we must use others cofactor. Don’t forget to obseve the mark : (+) or (-)
- 16. PROBLEM Find Adjoint of matrix Solution : OR
- 17. Adj or Adj
- 18. G. INVERSE OF MATRIX It is easy to show that the inverse of matrix is uniqe and the inverse of the inverse of A is A-1 but there is also many properties inverse matix; that is, a. − = the inverse of matrix = ( ) b. − = = () − For any nonsingular matrix A c. = = For any square matrix A d. − = If A is nonsingular e. = , = − If A is an m x n nonsingular matrix, = , = − If B is an n x m matrix, and there exist matrix X f. − = − − For any two nonsingular matrices A and B
- 19. A square matrix that has an inverse is called a nonsingular matrix A matrix that does not have an inverse is called a singular matrix Square matrices have inverses except when the determinant is zero When the determinant of a matrix is zero the matrix is singular
- 20. EXAMPLE 1 2 A= 3 4 1 1 4 2 0.4 0.2 A 10 3 1 0.3 0.1 To check AA-1 = A-1 A = I 1 1 2 0.4 0.2 1 0 AA I 3 4 0.3 0.1 0 1 1 0.4 0.2 1 2 1 0 A A I 0.3 0.1 3 4 0 1
- 21. Example 2 3 1 1 A 2 1 0 1 2 1 The determinant of A is |A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2 The elements of the cofactor matrix are c11 ( 1), c12 ( 2), c13 (3), c21 ( 1), c22 ( 4), c23 (7), c31 ( 1), c32 ( 2), c33 (5),
- 22. The cofactor matrix is therefore 1 2 3 C 1 4 7 1 2 5 so 1 1 1 adjA C T 2 4 2 3 7 5 and 1 1 1 0.5 0.5 0.5 1 adjA 1 A 2 4 2 1.0 2.0 1.0 A 2 3 7 5 1.5 3.5 2.5