# Math Help - Matrix problem

1. ## Matrix problem

If $A=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$, find $c,d$

so that $(c I + d A)^2 = A$

2. Originally Posted by flintstone
If $A=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$, find $c,d$

so that $(c I + d A)^2 = A$

I will assume that c and d are scalar constants, while I is the indentitiy matrix. Let's start out by doing some of the small arithmetic:

$cI = \begin{pmatrix}c & 0 \\ 0 & c\end{pmatrix}$

$dA = \begin{pmatrix}0 & d \\ -d & 0\end{pmatrix}$

$
cI + dA = \begin{pmatrix}c & d \\ -d & c\end{pmatrix}
$

$
(cI + dA)^2 = \begin{pmatrix}c & d \\ -d & c\end{pmatrix}
\begin{pmatrix}c & d \\ -d & c\end{pmatrix} =
\begin{pmatrix}c^2-d^2 & 2cd \\ -2cd & c^2-d^2\end{pmatrix}
$

Now you should be able to set your c's and d's so that the matrix looks all like original A again.

-Andy