Matrices A rectangular array of numbers like 2
Matrices A rectangular array of numbers like 2 1 3 A 1 0 2 is called a matrix. We call A a "2 by 3" matrix because it has two rows and three columns. An "m by n" matrix has m rows and n columns, for example a 11 a A 21 a m 1 a 12 a 22 a m 2 a 1 n a 2 n matrix a mn If m n Rectangular matrix If m n Square matrix If m 1 and n 1 Row matrix or row vector If m 1 and n 1 Column matrix or column vector Ex 1: 1 3 A 2 2 matrix ; Square matrix 0 7 B 4 1 3 1 3 matrix ; Row matrix 2 8 C 4 1 matrix ; Column matrix 5 1 2 1 3 1 D 2 4 matrix ; (Rectangular matrix) 1 8 0 2 Zero Matrix It is a matrix in which all the elements are zero, for example 0 0 O 2 2 zero matrix 0 0 -1 -
0 0 0 O 0 0 0 3 3 zero matrix 0 0 0 Identity Matrix It is a square matrix in which all the main diagonal elements have the value of one and the others are zero. 1 0 0 I 0 1 0 3 3 Identitymatrix 0 0 1 1 0 0 0 0 1 0 0 4 4 Identitymatrix I 0 0 1 0 0 0 Equality Two matrices are equal if (and only if) they have the same numbers in the same positions. For example b 11 B b 21 b 12 b 13 3 2 5 b 22 b 23 1 1 2 if and only if b 1 1 3, b 1 2 2 , b 1 3 5 , b 2 1 1, b 2 2 1, b 2 3 2. Transpose of a Matrix It is a matrix obtained when the rows of the original matrix are written as columns in the same order. T 3 1 3 2 6 If A 2 3 the transpose of A A T 2 3 1 3 5 6 5 -2 -
Matrix Addition and subtraction Two matrices with the same shape can be added (or subtracted) by adding (or subtracting) corresponding elements. Ex 2: If 3 1 8 A 2 3 6 7 4 1 and 6 B 4 9 1 3 2 1 5 2 Find C = A + B, and D = A – B. Solution 1 1 8 3 9 0 3 6 C 2 4 3 2 6 1 6 1 4 2 8 12 7 5 1 9 1 1 8 3 3 2 3 6 5 D 2 4 3 2 6 1 2 4 2 10 7 5 2 1 9 11 5 65 7 2 Matrix Multiplication (multiplication by constant) In general, if k is a scalar, then a 11 k a 21 a 3 1 a 12 a 22 a 32 a 13 k a 1 1 a 23 k a 21 a 3 3 k a 3 1 k a 1 2 k a 22 k a 32 k a 1 3 k a 23 k a 3 3 Ex 3: Given 2 1 A 1 3 and 1 3 B 3 4 Find C = 3 A- B. Solution 2 C 3 1 1 1 3 6 3 1 3 3 3 4 3 9 3 4 5 6 C 0 5 -3 -
Multiplication of two matrices Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. If A k m and B m n A B matrix of order k n a 11 If A a 21 a 12 a 22 and a 11 b 11 a 12 b 21 A B a 21 b 11 a 22 b 21 b B 11 b 21 b 12 , then : b 22 a 11 b 12 a 12 b 22 a 21 b 12 a 22 b 22 Ex 4: Find A B if: (1) 2 3 A 1 4 3 2 3 1 2 1 3 2 A B 1 1 4 2 3 1 1 2 8 A B 9 1 (2) 1 2 7 and 1 B 2 1 2 3 1 1 2 2 2 3 1 2 1 3 1 1 2 4 1 1 1 4 1 3 2 1 1 3 1 1 3 3 3 4 3 1 7 A 1 2 2 3 3 2 4 1 and 1 3 2 B 1 2 3 3 3 6 4 2 (H. W) Note A B B A Matrix Inverse If A B I , then B is the inverse matrix of A and denoted as A 1 , where: A 1 adj A det A (adj A = adjoint A) -4 -
a 11 If A a 21 a 3 1 a 12 a 22 a 3 2 A 11 A 12 adj. A 21 A A 3 1 A 22 A 32 a 13 a 23 , then a 3 3 A 13 T A 11 A 23 A 12 A 3 3 A 1 3 A 21 A 22 A 23 A 31 A 32 A 3 3 To find A 1 , A must be square matrix and det A 0. Ex 5: Find A 1 if: (1) 4 1 A 2 3 det A 4 1 2 3 A 11 adj. A A 21 12 2 10 T A 12 A 11 A 22 A 1 1 1 3 3; A 11 A 21 12 1 1 1; A A 21 A 22 1 1 2 22 1 2 2 4 4 T 3 2 3 adj. A 1 4 2 1 0. 3 4 0. 2 A 1 1 3 10 2 (2) 1 1 A 2 3 1 2 1 det A 2 1 1 3 2 1 4 0. 1 0. 4 2 1 1 2 1 1 3 2 1 2 4 3 8 1 -5 -
A 11 T A 11 A 13 A 23 A 12 A 13 A 3 3 A 12 adj. A A 21 A 31 A 11 1 A 22 A 32 3 1 1 1 5 , A 12 2 1 A 21 1 1 2 A 31 1 1 3 2 1 3 1 5 adj. A 3 7 2 2 1 A 22 A 32 7 , A 32 1 3 1 0 (3) A 5 2 1 1 4 2 2 2 1 3 1 1, 23 A 1 3, A 33 1 3 3 1 5 1 3 6 11 1 0 1 Ans. A 1 9 9 18 (H. W) 2 Solving System of Linear Equations by Matrices The following equations a 1 1 a b 2 x 21 a 31 a 1 2 x x a y 22 a 32 a 1 3 y y z a 23 a 33 b 1 z z b 3 can be represented in matrix form as follows: A X B , where x b 1 a 12 a 13 A a 21 a 22 a 23 , X y , and B b 2 z a 3 1 a 3 2 a 3 3 b 3 and the solution can be calculated from -6 - 2 1 7 3 3 3 2 3 7 3 5 1 7 5 1 3 1 8 5 7 1 1 3 2 T 2 1 1 2 2 1 7 5 1 1 3 5 7 5 1 1 A 1 8 7 A 31 A 23 A 3 3 2 1 2 1 3 1 1, A 13 1 1 3, A 22 1 A 21 7 1 1 1 2 1 1 2 3 1 5
x X y A 1 B z Ex 6: Solve the following equations by matrices x y 2 z 1 2 x 3 y z 14 x 2 y z 3 Solution 1 A 2 1 1 3 2 x 1 , X y , and B 14 1 z 3 2 1 det A 2 1 3 1 2 5 1 1 8 and A 1 1 8 1 7 2 5 x 1 X y A 1 B 1 8 z 7 7 1 3 1 5 3 7 1 16 2 1 1 3 14 24 3 8 1 5 3 8 1 3 x 2 , y 3, and z 1 H. W: Solve the following equations using matrices x y 2 z 1 (1) 2 x 3 y z 14 x 2 y z 3 (H. W) Ans. x = -1, y = 2, and z = 3 x 2 y z 4 (3) 3 x 5 y z 5 2 x y 2 z 5 (H. W) Ans. x = -2, y = 1, and z = 4 -7 -
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