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# Matrix algebra

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### Matrix algebra

1. 1. An introduction to Matrix Algebra
2. 2. Algebra
3. 3. MATRIX A matrix is an ordered rectangular array of numbers, arranged in rows and columns. rows columns
4. 4. ORDER OF A MATRIXThe size or order of a matrix isdescribed by its number of rowsand the number of columns. If a matrix, A, has m rows and n columns then A is described as an mxn matrix.
5. 5. The numbers in a matrix are called itselements. The element in the ith row and jthcolumn of a matrix is generally denoted byaij. A matrix with m rows and n columns iswritten or .
6. 6. Row Matrix A matrix with just one row is called a row matrix (or row vector). A a1 a 2 , an aj (1 x n)
7. 7. Column Matrix A matrix with just one column is called a column matrix. a1 a2 A ai (m x 1) am
8. 8. Matrices of the same orderTwo matrices which have the Samenumber of rows and columns aresaid to be matrices of the sameorder.
9. 9. Equal MatricesTwo matrices A = (aij) and B = (bij) are said to be equal if,and only if, each element aij of A is equal to thecorresponding element bij of B.In symbolic form this reads: A=B  aij = bij for all i and jFrom this it follows that equal matrices are of the sameorder but matrices of the same order are not necessarilyequal.
10. 10. Null matrix Any matrix, all of whose elements are zero, is called a null or zero matrix and is denoted by O.
11. 11. Matrix Addition A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by: cij aij bij Note: all three matrices are of the same dimension
12. 12. Addition a11 a12 If A a 21 a 22 b11 b12 and B b 21 b 22 a11 b11 a12 b12 then C a 21 b 21 a 22 b22
13. 13. Matrix Addition Example 3 4 1 2 4 6 A B C 5 6 3 4 8 10
14. 14. Multiplication by a scalar If A is a given matrix and a scalar then A is the matrix each of whose elements is times the corresponding element of A.Thus A
15. 15. TheIdentity
16. 16. Identity Matrix Square matrix with ones on the diagonal and zeros elsewhere. 1 0 0 0 0 1 0 0 I 0 0 1 0 0 0 0 1
17. 17. Equal Matrices Two matrices A and B are said to be equal if, and only if, each element aij of A is equal to the corresponding element bij of B.
18. 18. The Null matrix Any matrix all of whose elements are zero is called a null or zero matrix
19. 19. Transpose Matrix Rows become columns and columns become rows a11 a 21 , , am1 a12 a 22 , , am 2 A a1n a 2n , , amn
20. 20. Square Matrix Same number of rows and columns 5 4 7 B 3 6 1 2 1 3
21. 21. Matrix Subtraction C = A - B Is defined by Cij Aij Bij
22. 22. Matrix Multiplication Let A and B be two matrices. If the number of columns in A is equal to the number of rows in B we say that A and B are conformable for the matrix product AB. If A is order m n and B is of order n p, then the product AB is defined and is a matrix of order m p.
23. 23. Matrix Multiplication Matrices A and B have these dimensions: [r x c] and [s x d]
24. 24. Matrix Multiplication Matrices A and B can be multiplied if: [m x n] and [n x p] n=n
25. 25. Matrix MultiplicationThe resulting matrix will have the dimensions: [m x n] and [n x p] mxp
26. 26. Computation: A x B = C a11 a12 A a 21 a 22 [2 x 2] b11 b12 b13 B b 21 b 22 b 23 [2 x 3] a11b11 a12b21 a11b12 a12b22 a11b13 a12b23 C a 21b11 a 22b21 a 21b12 a 22b22 a 21b13 a 22b23 [2 x 3]
27. 27. Computation: A x B = C 2 3 111 A 11 and B 1 0 2 1 0 [3 x 2] [2 x 3] A and B can be multiplied 2 *1 3 *1 5 2 *1 3 * 0 2 2 *1 3 * 2 8 528 C 1*1 1*1 2 1*1 1* 0 1 1*1 1* 2 3 213 1*1 0 *1 1 1*1 0 * 0 1 1*1 0 * 2 1 111 [3 x 3]
28. 28. Computation: A x B = C 2 3 111 A 11 and B 1 0 2 1 0 [3 x 2] [2 x 3] Result is 3 x 3 2 *1 3 *1 5 2 *1 3 * 0 2 2 *1 3 * 2 8 528 C 1*1 1*1 2 1*1 1* 0 1 1*1 1* 2 3 213 1*1 0 *1 1 1*1 0 * 0 1 1*1 0 * 2 1 111 [3 x 3]
29. 29. Note: If A is an m n and B is n p matrix, then AB is an m p matrix. Hence we see that BA is defined only when p=m.
30. 30. Inversion
31. 31. The Inverse of a MatrixDefinition:Let A be a square matrix. A matrix Bsuch that AB=I=BA is called the inversematrix of A and is denoted by A-1.So if A-1 exists, we have AA-1=I=A-1Aand the matrix is said to be invertible.If a matrix has no inverse, then it is saidto be non-invertible.
32. 32. The Inverse of a Matrix 1 1 A A AA I Like a reciprocal Like the number one in scalar math in scalar math
33. 33. Linear System of SimultaneousEquations First precinct: 6 arrests last week equally divided between felonies and misdemeanors. Second precinct: 9 arrests - there were twice as many felonies as the first precinct. 1st Precinct : x1 x2 6 2nd Pr ecinct : 2x1 x2 9
34. 34. 11 11Solution Note: Inverse of 21 is 2 1 11 x1 6 * 21 x2 9 11 11 x1 11 6 Premultiply both sides by * * * inverse matrix2 1 21 x2 2 1 9 10 x1 3 A square matrix multiplied by its * inverse results in the identity matrix. 01 x2 3 x1 3 A 2x2 identity matrix multiplied by the 2x1 matrix results in the original x2 3 2x1 matrix.
35. 35. General Form n equations in n variables: n aijxj bi or Ax b j 1 unknown values of x can be found using the inverse of matrix A such that 1 1 x A Ax A b
36. 36. Garin-Lowry Model Ax y x The object is to find x given A and y . This is done by solving for x : y Ix Ax y (I A)x 1 (I A) y x
37. 37. Matrix Operations in Excel Select the cells in which the answer will appear
38. 38. Matrix Multiplication in Excel 1) Enter “=mmult(“ 2) Select the cells of the first matrix 3) Enter comma “,” 4) Select the cells of the second matrix 5) Enter “)”
39. 39. Matrix Multiplication in Excel Enter these three key strokes at the same time: control shift enter
40. 40. Matrix Inversion in Excel Follow the same procedure Select cells in which answer is to be displayed Enter the formula: =minverse( Select the cells containing the matrix to be inverted Close parenthesis – type “)” Press three keys: Control, shift, enter