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Section 9.5 The Algebra of Matrices Objectives: • To understand matrix addition and subtraction. • To understand scalar and matrix multiplication. Equality of Matrices Two matrices are equal if they have the same entries in the same positions. Equal Matrices 4 22 e 0 2 4 1 1 2 0 0.5 1 1 1 2 2 Unequal Matrices 1 2 3 4 1 3 5 2 4 6 5 6 Ex 1. Determine if the matrices are equal. 2 7 0 2 1 5 0 30 14 2 5 Addition, Subtraction, and Scalar Multiplication • Two matrices can be added or subtracted if they have the same dimension. – Otherwise, their sum or difference is undefined. – We add or subtract the matrices by adding or subtracting corresponding entries. Scalar Multiplication • To multiply a matrix by a number, we multiply every element of the matrix by that number. Ex 2. Carry out each indicated operation, or explain why it cannot be performed. (a) A + B (b) C – D (c) C + A (d) 5A 2 3 A 0 5 7 21 7 3 0 C 0 1 5 1 B 3 2 6 D 8 0 1 2 0 6 1 9 Class Work Determine if the matrices are equal. 1 1. 5 3 2 5 8 log 4 4 2 .5 4 8 7 5 A 1 12 B 4 20 7 9 8 11 Find. 2.A + B 3.2A 4.3A - B HW p684 1-6 all, 17-23 odd Class Work 1. Write the system as an augmented matrix and then solve. x 9 y 7 z 41 8 x 2 y 6 z 70 3 x 5 y 4 z 27 Perform the indicated operation or explain why it is not possible. 9 19 11 3 6 6 P Q 4 2 13 18 11 2 2. Q – P 3. P + R 4. 6R 3 6 R 4 7 11 5 Matrix Multiplication If A is an m x n matrix and B is an n x k matrix then AB has dimension m x k. That is, two matrices can be multiplied if and only if the number of columns in A matches the number of rows in B. Ex 3. Find AB. 1 3 1 5 2 A B 1 0 0 4 7 Ex 4. Find CD. 4 1 1 2 C 3 0 D 3 1 2 1 Class Work 5. Find DC. 4 1 1 2 C 3 0 D 3 1 2 1 5 7 1 2 A B 3 0 9 1 6. Find AB 7. Find BA. Writing a system as a Matrix Equation A matrix equation is in the form AX = C where A is a coefficent matrix, X is a variable matrix, and C is the constant matrix. Ex 5. Write the system as a matrix equation. x y 3z 4 x 2y 2z 10 3 x y 5z 14 Class Work 8. Write the system as a matrix equation. x y z 6 4 x 6y 8z 20 2x 5y 3z 19 HW p685 18, 20, 22, 25, 27, 28, 31, 35, 43, 46