Linear Algebra: A Modern Introduction

Published by Cengage Learning
ISBN 10: 1285463242
ISBN 13: 978-1-28546-324-7

Chapter 3 - Matrices - 3.2 Matrix Algebra - Exercises 3.2 - Page 161: 5

Answer

$$B=2A_1+A_2.$$

Work Step by Step

Let $$ B=\left[\begin{array}{ccc}{2} & {5} \\ {0} & {3}\end{array}\right], \quad A_1=\left[\begin{array}{ccc}{1} & {2} \\ {-1} & {1}\end{array}\right], \quad A_2=\left[\begin{array}{ccc}{0} & {1} \\ {2} & {1}\end{array}\right] $$ We have $$B=aA_1+bA_2=a\left[\begin{array}{ccc}{1} & {2} \\ {-1} & {1}\end{array}\right]+b\left[\begin{array}{ccc}{0} & {1} \\ {2} & {1}\end{array}\right]$$ and hence we get the system $$a=2, \quad 2a+b=5,\quad -a+2b=0, \quad a+b=3.$$ The above system has the solution $a=2$ and $b=1$, so $$B=2A_1+A_2.$$
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