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Found in: Page 308

### Linear Algebra With Applications

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

# Find all ${\mathbf{2}}{\mathbf{×}}{\mathbf{2}}$ matrices A such that ${\mathbf{adj}}\left(A\right){\mathbf{=}}{{\mathbit{A}}}^{{\mathbf{T}}}$.

Therefore, the $\mathrm{adj}\left(A\right)={A}^{T}$ is given by,

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

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 “.”

## Step 2: To find adj(A)=AT.

Let,

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

Then, we have

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

and

$\mathrm{adj}A=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

For ${A}^{T}=\mathrm{adj}A$, we need $a=d$ and $b=-c$.

Thus,

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