# Math Help - Help getting started on a problem (intro linear algebra)

1. ## Help getting started on a problem (intro linear algebra)

Q: Suppose $A=\begin{bmatrix}a&b\\c&d&\end{bmatrix}$. Find conditions on $a,b,c,$and $d$ for which $A^{-1}=A$. Then find two different matrices, other than the identity matrix, for which $A^{-1}=A$.

Do I just set up an equation to find what $A^{-1}$ is and then equate that to $A$ to find out when $A^{-1}=A$?

Or could I use the face that if $A^{-1}=A \Rightarrow A^{2}=I$?

Im not really sure to I pull everything together.

Thanks

2. You are correct. If $A=A^{-1}$, then $A^{2}=I$

$\begin{bmatrix}a&b\\c&d\end{bmatrix}$ $\cdot \begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatri x}1&0\\0&1\end{bmatrix}$

$\begin{bmatrix}a^{2}+bc&ab+bd\\ac+cd&bc+d^{2}\end{ bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$

$a^{2}+bc=1$
$ab+bd=0$
$ac+cd=0$
$bc+d^{2}=1$

One such matrix that fits is $\begin{bmatrix}1&0\\-3&-1\end{bmatrix}$