Presentation On Matrices Mathematics DIU
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The document provides an overview of matrices, including their definitions, types (such as row, column, rectangular, square, null, diagonal, scalar, negative, and identity matrices), and various mathematical operations related to them. It explains the elements, order of matrices, and concepts like transpose and adjoint. Additionally, the document illustrates these concepts with examples to further clarify the definitions.
Presentation On Matrices Mathematics DIU
- 2. Introduction: Joseph James Sylvester was a English mathematician (1814-1897) used the word matrix first time. Matrix: Matrix is singular while matrices is plural. Definition: A matrix is a rectangular array of number, symbols or expressions arranged in rows and columns. For Examples. A= 2 3 5 6 7 9 11 8 11 C= 9 D= 2 9 6 8 E= 5 7 9
- 3. Definition: A Matrix is a bracket containing an array of numbers or letters arranged in rows and in columns. For Examples: A= 5 7 6 6 9 7 4 2 4 5 6 3 C= 7 1 6 4 6 8 Elements Or Entries Of Matrix. Each numbers or letters used in a matrix is called elements or entries of matrix. Order Of Matrix. The Number of rows by the number of columns is called order of matrix. It is written as m × n ( Here it is not product that you multiply here this symbol means m by n)
- 4. m n is called order of the matrix. We usually use capital latters such as A, B, C, D, etc. to represent the matrices and Small letters such as a, b, c, d,etc. to indicate the entries of the matrices. Note: The elements(entries) of matrices need not always be numbers but in the study of matrices, we shall take the elements of the matrices from Real numbers or complex numbers. Kinds Of Matrix. Row Matrix. A Matrix having only one row is called row matrix. A matrix having only one row with two or more columns is known as row matrix. A= 2 4 5 6 8 is a row matrix of order 1× 5
- 5. Column Matrix. A Matrix having only one column is called column matrix. A matrix having only one column with two or more rows is known as column matrix. B = 6 3 8 isa column matrixof order 1× 3 Rectangular Matrix. If in a matrix the number of rows and the number of columns are not equal then the matrix is called rectangular matrix. If m ≠ n, (If m is not equals to n), then the matrix is called a rectangular matrix of order m × n. that is, the matrix in which the number of rows is not equal to the number of columns is said to be rectangular matrix.
- 6. For Example. 3 6 C= 9 11 is a rectangular matrix of order 3 ×2 4 2 Square Matrix. If matrix in which number of rows and number of columns are equal then the matrix called square matrix. If m=n, then the matrix of order m ×n is said to be square matrix of order n or m. i.e., the matrix which has the same number of rows and columns is called a square matrix. For Example. C= 3 isa square matrix. 4 6 3 A= 5 4 6 isa square matrix of order 3 × 3 . 3 4 2
- 7. Null MatrixOr Zero Matrix. A matrixwhose each element iszero iscalled a null or zero matrix. it isdenoted by “0”. For Example. A=0 0 0 0 isnull matrixof order 0. Diagonal Matrix. A square matrix in which all the elements except at least one element of diagonal are zero. A square matrix in which all its elements are zero except the diagonal which runs from upper left to lower right is known as diagonal matrix. For Example. A= 4 0 0 4 1 0 0 2 0 0 B= 0 1 0 C= 0 4 0 0 0 1 0 0 2
- 8. ScalarMatrix. A diagonal matrix having equal elements in its diagonal is called scalar matrix. For Example. C= 4 0 0 4 2 0 0 A= 0 2 0 0 0 2 Negative Matrix. If signs of all the entries of a matrix A changed the new matrix obtained will be negative matrix of matrix A is called negative matrix. It isdenoted by –A. For Example. -5 4 -3 5 -4 3 A= 6 8 1 -A= -6 -8 -1 is a negative matrix of A matrix. 3 9 5 -3 -9 -5
- 9. Unit Matrixor Identity Matrix. A scalar matrix having each elements of a diagonal equal to 1 is called Unit Matrix. It isDenoted By Capital latter I. For Example. 1 0 0 1 1 0 0 A= 0 1 0 C= 0 0 1 T ranspose Of a Matrix. If A is a matrix of order m × n then a matrix of order n ×m obtained by interchanging the rows and columns of A matrixiscalled T ranspose of A Matrix. It isdenoted by 𝐴𝑡 .
- 10. 3 4 2 3 4 7 A= 4 3 1 𝐴𝑡= 4 3 2 7 2 8 2 1 8 Adjoint of a Matrix. Let A= 𝑎 𝑏 𝑐 𝑑 Isa matrix. Then the matrix obtained by interchanging the elements of primary diagonal, i.e. a and d and by changing the sings of the other elements, i.e. b and c. It isdenoted by adj A. adj A= 𝑑 −𝑏 −𝑐 𝑎
- 11. Primary and Secondary Diagonal of a matrix. A= 1 1 is primary diagonal. 0 P=3 2 5 1 Q= 1 2 3 4 Solution. P+Q= 3 2 5 1 + 1 2 3 4 0 is secondary diagonal 1 0 0 1 Addition of Matrices. If P and Q are two matrices of the same order then the sum is obtained by adding their corresponding elements. The sum of P and Q is denoted by P+Q and its order will be equal to the order of matrix P and Q For Example. Let. P and Q
- 12. P+Q= 3 + 1 2 + 2 5 + 3 1 + 4 P+Q= 4 4 8 5