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Matrix Algebra seminar ppt
- 1. MATRIX ALGEBRA MADE BY:- Swetalina pradhan Priyanka panigrahi
- 2. INDEX 1.MATRIX I . Order of matrix 2.TYPES OF MATRIX I . Column matrix II . Row matrix III . Square matrix IV . Diagonal matrix V. Identity matrix VI . Null matrix 3.OPERATIONS ON MATRIX I . Addition and subtraction II . Multiplication 4.TRANSPOSE OF A MATRIX 5.SYMMETRIC AND SKEW SYMMETRIC MATRIX 6.INVERTIBLE MATRIX 7.APPLICATION OF MATRIX
- 3. MATRICES A matrix is a structural representation of rows and columns which is enclosed within two brackets. dc ba 03 24 11 The horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix. 2x2 2x2 1x2
- 4. ORDER OF MATRICES mnmm n n aaa aaa aaa 21 22221 11211 ... ... A matrix having m rows & n columns is called a matrix of order m × n or simply m × n matrix General representation of Matrices m × n is
- 5. TYPES OF MATRICES 1. Column matrix or vector: 2 4 1 3 1 1 21 11 ma a a A matrix is said to be a column matrix if it has only one column. In general, A = [aij] m × 1 is a column matrix of order m × 1. 2x13x1
- 6. 611 2530 naaaa 1131211 2. Row matrix or vector: A matrix is said to be a column matrix if it has only one Row. In general, A = [aij] 1 × n is a column matrix of order 1 x n. 1x3 1x4
- 7. A matrix, in which the number of rows is equal to the number of columns, is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. 3. Square matrix 03 11 166 099 111 2x2 3x3
- 8. 4. Diagonal matrix A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non diagonal elements are zero. That is a matrix B = [bij] m × m is said to be a diagonal matrix if bij = 0, when i ≠ j. 100 020 001 9000 0500 0030 0003 3x3 4x4
- 9. 5. Unit or Identity matrix A square matrix in which elements in the diagonal are all 1 and rest is all zero is called an identity matrix. 1000 0100 0010 0001 10 01 2x2 4x4
- 10. 6. Null (zero) matrix A matrix is said to be zero matrix or null matrix if all its elements are zero. We denote zero matrix by O. 0 0 0 000 000 000 3x1 3x3
- 11. EQUALITY OF MATRICES Two matrices are said to be equal only when all corresponding elements are equal Therefore their size or dimensions are equal as well. A = 325 012 001 B = 325 012 001 A = B 3x3 3x3
- 12. OPERATIONS ON MATRICES ADDITION AND SUBTRACTION OF MATRICES The sum or difference of two matrices, A and B of the same size yields a matrix C of the same size. ijijij bac Matrices of different sizes cannot be added or subtracted
- 14. PROPERTIES OF MATRIX ADDITION Commutative Law: A + B = B + A Associative Law: A + (B + C) = (A + B) + C = A + B + C Existence of additive identity A + 0 = 0 + A = A The existence of additive inverse A + (-A) = 0
- 15. SCALAR MULTIPLICATION OF MATRICES Matrices can be multiplied by a scalar (constant or single element) Let k be a scalar quantity; then kA = Ak 416 128 48 412 4 14 32 12 13 14 32 12 13 4 4x24x2 4x2
- 16. MULTIPLICATION OF MATRICES The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. )37()22()84()57()62()44( )33()22()81()53()62()41( 35 26 84 724 321 5763 2131 2x2 3x2 2x3 2x2
- 17. Remember also: IA = A 10 01 5763 2131 5763 2131 AB not generally equal to BA. 010 623 05 21 20 43 2015 83 20 43 05 21 20 43 05 21 ST TS S T 2x22x22x2 2x2 2x2 2x2 2x2
- 18. TRANSPOSE OF A MATRIX If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A′. 135 742 A Then transpose of A, denoted AT is: 17 34 52 T A 2x3 3x2
- 19. SYMMETRIC MATRICES A Square matrix is symmetric if it is equal to its transpose: A = AT 741 45.13 132 A 741 45.13 132 A 3x3 3x3
- 20. SKEW SYMMETRIC MATRICES A square matrix A is said to be a skew symmetric if B’ = – B . And all elements in the principal diagonal of a skew symmetric matrix are zeroes. 0 0 0 gf ge fe B 3x3
- 21. IMPORTANT POINT i. For every square matrix A, A+A' is a symmetric matrix and A+A' is a skew symmetric matrix. ii. Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix Let A be a square matrix, then we can write A=1/2(A+A')+1/2(A-A')
- 22. INVERTIBLE MATRICES For example:- A square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I = BA, Where I is identify matrix of order n. 12 25 B 52 21 A 2x2 2x2
- 23. 10 01 12 25 52 21 AB 10 01 52 21 12 25 BA 2x2 2x2 2x2 2x2 2x2 2x2 Thus, B is the inverse of A, in other words B=A– 1 and A is inverse of B .
- 24. Application of matrix Solving linear equations -using inverse matrix Computer graphics
- 25. ANY QUESTIONS