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Q28E

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Found in: Page 309

### Linear Algebra With Applications

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

# If the determinant of a ${\mathbf{2}}{\mathbf{×}}{\mathbf{2}}$ matrix is 4, then the inequality $|\left|A\stackrel{\to }{v}\right||{\mathbf{\le }}{\mathbf{4}}|\left|\stackrel{\to }{v}\right||$ must hold for all vectors $\stackrel{\mathbf{\to }}{\mathbf{v}}$ in .

Therefore, the given statement 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: To check whether the given condition is true or false

Let,

$A=\left[\begin{array}{cc}4& 4\\ 0& 1\end{array}\right]$.

Clearly,

det A = 4 , however, for

$V=\left[\begin{array}{c}1\\ 1\end{array}\right]$

We have,

$\left|\left|Av\right|\right|=\left|\left|\left[\left[\begin{array}{c}8\\ 1\end{array}\right]\right]\right|\right|\phantom{\rule{0ex}{0ex}}\left|\left|Av\right|\right|=\sqrt{65}>\sqrt{32}\phantom{\rule{0ex}{0ex}}\left|\left|Av\right|\right|=4\left|\left|v\right|\right|$

Therefore, the given statement is true.

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