Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 10 - Systems of Equations and Inequalities - Section 10.4 Matrix Algebra - 10.4 Assess Your Understanding - Page 777: 61


The matrix $A$ has no inverse.

Work Step by Step

In order to calculate the inverse of an $n$ by $n$ non-singular matrix $A$, we will proceed with the following steps: Step 1. Transform the given matrix $A$ into this form $\left[A|I_{n}\right]$ as follows: $\left[A|I_{n}\right]$ = $\left[\begin{array}{ll|ll} {4}&{2}&{1}&{0}\\ {2}&{1}&{0}&{1}\end{array}\right]$ Step 2. Transform the matrix $\left[A|I_{n}\right]$ into reduced row-echelon form by using the row operations: $R_{2}=-\dfrac{r_{1}}{2}+r_{2}$ $\left[\begin{array}{rr|rr}{4}&{2}&{1}&{0}\\ {0}&{0}&{-\dfrac{1}{2}}&{1}\end{array}\right]$ We see that the zeros on the left of the vertical line in Row-2 make it impossible to obtain a leading nonzero entry in Column-2. This means that the identity matrix does not occur on the left of the reduced row-echelon form of $\left[A|I_{n}\right]$, so the matrix $A$ has no inverse.
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