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Expert-verified Found in: Page 263 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # TRUE OR FALSE? The equation ${\mathbit{d}}{\mathbit{e}}{\mathbit{t}}\left({A}^{T}\right){\mathbf{=}}{\mathbit{d}}{\mathbit{e}}{\mathbit{t}}\left(A\right)$ holds for all ${\mathbf{2}}{\mathbf{×}}{\mathbf{2}}$ matrices A.

The given statement is true.

See the step by step solution

## Step 1: Definition of the determinant.

Suppose ${\mathbit{A}}{\mathbf{=}}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is a ${\mathbf{2}}{\mathbf{×}}{\mathbf{2}}$ matrix.

Then the determinant of matrix A is defined as,

${\mathbit{d}}{\mathbit{e}}{\mathbit{t}}\left(A\right){\mathbf{=}}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\phantom{\rule{0ex}{0ex}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{}}{\mathbf{=}}{\mathbit{a}}{\mathbit{d}}{\mathbf{-}}{\mathbit{b}}{\mathbit{c}}$

## Step 2: Check whether the given statement is a true or false.

The given statement is the equation $det{\left(A\right)}^{T}=det\left(A\right)$ holds for all $2×2$matrix A.

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$.

Then,

${A}^{T}={\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{T}\phantom{\rule{0ex}{0ex}}{A}^{T}=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]$

Find $det\left({A}^{T}\right),det\left(A\right)$

$det\left(A\right)=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\phantom{\rule{0ex}{0ex}}=ad-bc\phantom{\rule{0ex}{0ex}}det\left({A}^{T}\right)=\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|c\phantom{\rule{0ex}{0ex}}=ad-cd$

This is clear that, the equation $det\left({A}^{T}\right)=det\left(A\right)$is holds for all $2×2$ matrix A

Then the given statement is true. ### Want to see more solutions like these? 