4.0 MATRICES AND SYSTEMS OF LINEAR EQUATIONS LECTURE 1 OF 5 4.1 MATRICES
LEARNING OUTCOMES At the end of this topic, students should be able to : (a) Identify the different types of matrices. (b) Perform operations on matrices. (c) Find the transpose of a matrix.
MATRICES A matrix is a rectangular array of numbers enclosed between brackets. The general form of a matrix with m rows and n columns : 11 12 13 1 21 22 23 2 31 32 33 3 1 2 3 n n n m m m mn a a a a a a a a A a a a a a a a a n columns m rows
• The numbers that makes up a matrix are called its entries or elements, aij , where i indicates the row and j indicates the column. 1 3 1 8 A a and a 12 22 3 8 Example: Given
• The order or dimension of a matrix with m rows and n columns is m x n. Example: order: 2x1 order: 2x2 1 8 0 1 3 4
1. Row Matrix is a (1 x n) matrix [one row] A = [ a11 a12 a13 … a1n] Example A = [ 1 2 ] B = [ 1 0 7 8 ]
11 21 31 1 m a a a . . . a 0 4 2. Column Matrix is a (m x 1) matrix [ one column ] A = Example B = A = 7 5 3 2
3. Square Matrix is a (nxn) matrix which has the same number of rows as columns. Example 2 3 1 3 1 2 1 3 2 1 8 1 3 A = 2 x 2 matrix B = 3 x 3 matrix
4. Zero Matrix is a (m x n) matrix which every entry is zero, and denoted by . Example 0 0 0 0 O = O = 0 0 0 0 0 0 O
5. Diagonal Matrix is a square matrix where all the elements are zero except those in leading diagonal Let A = 11 22 33 0 0 0 0 0 0 0 0 0 0 0 mm a a a a The diagonal entries of A are a11, a22 , ..., amm
A = 0 3 2 0 0 0 3 0 2 0 1 0 0 B = Example Diagonal Matrices
6. Identity Matrix is a diagonal matrix where all its diagonal entries are 1 and denoted by I. 0 1 1 0 0 0 1 0 1 0 1 0 0 I2x2 = I3x3 = Example
7. Lower Triangular Matrix is a square matrix and aij = 0 for i < j 3 2 3 3 2 0 1 0 0 c d e b f a 0 0 0 A = B = Example
8.Upper Triangular Matrix is a square matrix and aij = 0 for i > j 0 0 3 0 2 4 1 2 3 f d e a b c 0 0 P = R = 0 Example
The addition or subtraction of two matrices is only defined when they have the same order. Addition And Subtraction Of Matrices Operations on Matrices
3 4 1 2 A , 5 6 4 3 B . Find : (a) A + B (b) B - A Example 1
(a) A + B 3 4 1 2 + 5 6 4 3 1 4 2 3 3 ( 5) 4 6 = 5 5 2 10 Solution =
3 4 _ 1 2 b) B - A 4 3 5 6 3 1 8 2 =
Then . [ ] ij A a [ ] ij cA ca If c is a scalar and Scalar Multiplication 3 4 1 2 A , 5 6 4 3 B . Find 3A – 2B. Example 2
1 2 3 3 4 4 3 2 5 6 3 6 8 6 9 12 10 12 5 0 19 0 = = = Solution 3 2 A B
1. A + B = B + A (Commutative) 2. (A + B) + C = A + (B + C) (Associativite) 3. A + (-A) = (-A) + A = O (O, zero matrix) 4. ( ) A A A 5.(AB) AB 6.(A) ()A , constant Properties of Matrices
Multiplication of Matrices The product of two matrices A and B is defined only when the number of columns in A is equal to the number of rows in B. • If order of A is m x n and the order of B is n x p, then AB has order m x p. • AB≠BA
m x p is the order of the product AB These numbers must be equal m × n n × p Amxn Bnxp ABmxp
Multiplication Of Two Matrices n R a a a a 1 2 3 bn b b b C 3 2 1 1 1 2 2 3 3 [ ] RC a b a b a b a b n n
Let find AB. 0 2 5 1 3 , 2 A and B Example 3 Solution AB 2 0 5 3 1 2 17 A1x3 B3x1 AB1x1
Let and Find AB. 1 2 1 3 4 5 2 3 1 A 2 1 2 3 2 1 1 4 1 B Example 4
1 2 1 3 4 5 2 3 1 AB 2 3 (1) ( 1) 2 (4) (2) 1 ( 1)2 3 (1) ( 1) 2 (4 1 2 1 1 2 1 1 2 13 4 5 3 4 5 3 4 52 ) (2) 1 ( 1)2 3 (1) 3 1 2 3 ( 1) 2 (4) (2) 1 (1 2 3 11) 5 9 1 23 25 5 14 8 6 Solution 2 1 2 3 2 1 1 4 1
Definition The transpose of a matrix A , written as AT , is the matrix obtained by interchanging the rows and columns of A. That is, the i th column of AT is the i th row of A for all i’s. Transpose Matrix
1 3 5 2 5 4 1 3 3 If D then 3 1 3 1 2 B T If then B 1 3 213 Example 5 1 2 1 3 5 3 345 T D
If , and Find A - BT . A 2 5 4 3 1 1 B 2 5 3 3 2 4 Example 6
Solution T T A B 2 5 2 5 3 4 3 3 2 4 1 1 2 5 2 3 4 3 5 2 1 1 3 4 0 2 1 1 2 5
(A T ) T = A (A ± B) T = A T ± BT (kA) T = kAT , k is a scalar (AB) T = B TA T Properties of transpose
Let , and Verify that (BC) T = C TB T . 3 4 2 1 B 1 4 3 2 C Example 7
BC = 3 4 1 4 2 1 3 2 Solution = 2 3 8 2 3 12 12 8 (BC) T= 5 10 15 20 T = 5 10 15 20 = 20 10 15 5
(BC) T = CTBT C T B T = 1 3 3 2 4 2 4 1 = 12 8 8 2 3 12 2 3 = 20 10 15 5 = (BC) T
CONCLUSION MATRICES Transpose Operation Types of matrices -Row -Column -Square -Diagonal -Zero -Identity -Lower triangle -Upper triangle -Addition -Subtraction -Multiplication (A T ) T = A (A ± B) T = A T ± BT (kA) T = kAT (AB) T = B TA T