1. Unit 3 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 1 Unit 3 Matrix Arithmetic

2. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 2 3 Matrix Arithmetic 3.1 Introduction A matrix is a set of real or complex numbers (or elements) arranged in rows and columns to form a rectangular array. A matrix having m rows and n columns is an m x n (read ‘m by n’or ‘m cross n’) matrix and is referred to as having order m x n. A matrix can be represented explicitly by enclosing the array within large square brackets.

3. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 3 Example 3.1-11 1 3 4 6 9 is a 2 x 3 matrix, where 1, 1, 3, 4, 6, 9 are the elements of the matrix. Example 3.1-2 1 2 2 2 3 5 is a 4 x 3 matrix. 3 2 7 1 2 6

4. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 4 Example 3.1-35 2 7 1is a 1 x 4 matrix, (a row vector). Similarly,3 is a 2 x 1 matrix, 8(a column vector).

5. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 5 3.2Matrix notation Each element in a matrix has its own particular ‘address’ or location which can be defined by a system of double suffixes: the first indicates the row, and the second indicates the column, thus a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44

6. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 6 Further, a whole matrix can be denoted by a single general element enclosed in square brackets, or by a single letter printed in bold type. Example 3.2-1a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 can be denoted by [a ij ] or by A. Similarly, x 1 x 2 can be denoted by [x i ] or by X. x 3

7. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 7 3.3Equal Matrices Two matrices are said to be equal if all the corresponding elements are equal. Therefore, the two matrices must also be of the same order. Example 3.3-1

8. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 8 3.4 Addition and Subtraction of Matrices Only matrices of the same order can be added or subtracted. The result from the sum or difference is then determined by adding or subtracting the corresponding elements. Example3.4-1

9. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 9 3.5 Multiplication of Matrices Scalar multiplication Example 3.5-1 i.e. k[a ij ] = [ka ij ] This also means that we can take a common factor out of every element.

10. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 10 Multiplication of two matrices Two matrices can be multiplied together only when the number of columns in the matrix on the left equals the number of rows in the matrix on the right.

11. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 11 Example 3.5-2

12. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 12 Matrix multiplication is not commutative Example 3.5-3 5 2 A = 7 4 and B = 9 2 4 3 1-2 3 6 41 16 32 then AB = 55 52 52 and BA = 71 30 25 9 1829 14

13. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 13 3.6 Transpose of a Matrix If M is a matrix, its transpose is denoted by M T. 4 6 4 8 5 M = 8 3, then M T = 6 3 1 5 1

14. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 14 3.7 Special Matrix 3.7.1A square matrix is one in which number of rows = number of columns Example 3.7-1 Square matrixNot a square matrix

15. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 15 3.7.2Symmetric matrix A square matrix M is symmetric if M = M T Example 3.7-2 125125 M =289 M T = 289 594594

16. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 16 3.7.3Diagonal matrix A diagonal matrix M is a square matrix such that all of the off- diagonal elements are equal to 0. Example 3.7-3 100 M = 0-50 002 diagonal

17. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 17 3.7.4Unit matrix A unit matrix I is a square matrix with all elements in the diagonal equal to 1 and all off-diagonal elements equal to 0. Example 3.7-4 100 I = 010 001 Unit matrix behaves like the unit factor in ordinary algebra where IM = MI = M

18. Unit 03 Matrix Arithmetic IT Disicipline ITD 1111 Discrete Mathematics & Statistics STDTLP 18 3.7.5Null matrix A null matrix N is a matrix with all its elements equal to 0. Example 3.7-5 000 N = 000 000