Question

Find all $\mathbf{2}\mathbf{\times }\mathbf{2}$ matrices Asuch that$\mathbf{adj}\left(A\right)\mathbf{=}{\mathit{A}}^{\mathbf{T}}$.

Short answer

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

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

Step-by-step solution

Matrix Definition. 

Matrix is a set of numbersarranged in rows and columns soas to form a rectangulararray.

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 byn” matrix, written “.”

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,

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