Find the inverse, A-1, of the following matrix, in terms of the method of row reductions:3A11||

Answered: Find the inverse, A-1, of the following…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter6: Linear Systems

Section6.CR: Chapter Review

Problem 43E

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Question

Find the inverse, A-1, of the following matrix, in terms of the method of row reductions:
3
A
1
1
||

Expert Solution

Step 1

To obtain the inverse of the matrix A, apply the elementary row operations on the augmented matrix $[A | I]$ , to convert it in the form of $[I | A^{- 1}]$ .

To convert the first column entries, divide the first row by 3 and then subtract first row from third row.

$\begin{array}{rcl} [A | I] & = & [\begin{matrix} 3 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & - 1 & 0 & 0 & 0 & 1 \end{matrix}] \\ ~ [\begin{matrix} 1 & 0 & \frac{1}{3} & \frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & - 1 & 0 & 0 & 0 & 1 \end{matrix}] R_{1} & \to & \frac{R_{1}}{3} \\ ~ [\begin{matrix} 1 & 0 & \frac{1}{3} & \frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & - 1 & - \frac{1}{3} & - \frac{1}{3} & 0 & 1 \end{matrix}] R_{3} & \to & R_{3} - R_{1} \end{array}$

Step 2

To convert the second column entries, add third row and second row.

$\begin{array}{rcl} [A | I] ~ [\begin{matrix} 1 & 0 & \frac{1}{3} & \frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & - 1 & - \frac{1}{3} & - \frac{1}{3} & 0 & 1 \end{matrix}] \\ ~ [\begin{matrix} 1 & 0 & \frac{1}{3} & \frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & - \frac{1}{3} & - \frac{1}{3} & 1 & 1 \end{matrix}] R_{3} & \to & R_{3} + R_{2} \end{array}$

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