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Q11E

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Found in: Page 308

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# The matrix $\left[\begin{array}{ccc}{k}^{2}& 1& 4\\ k& -1& -2\\ 1& 1& 1\end{array}\right]$ is invertible for all positive constants k.

Therefore, the given condition is true.

See the step by step solution

## Step 1: Matrix Definition.

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “ m by n ” matrix, written “ $m×n$ .”

## Step 2: Given.

Given Matrix,

$\left[\begin{array}{ccc}{\mathrm{k}}^{2}& 1& 4\\ \mathrm{k}& -1& -2\\ 1& 1& 1\end{array}\right]$

## Step 3: To check whether the given condition is true or false.

We compute,

$\begin{array}{l}detA=-{k}^{2}-2+4k+4+2{k}^{2}-k\\ detA={k}^{2}+3k+2\end{array}$

which turns out to be 0 if and only if $k=-2$ or $k=-1$ , so for all positive constants $k$ , the matrix is regular.

Therefore, the given condition is true.