# Thread: Help getting started on a problem (intro linear algebra)

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

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

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

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

Im not really sure to I pull everything together.

Thanks

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

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

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

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

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