# Math Help - Matrices!

1. ## Matrices!

How do I do

If A = $\begin{bmatrix} 3 & 2 \\-2 & -1 \end{bmatrix}$, write A^2 in the form pA+qI where p and q are scalars?

How do I convert a square matrix to _A+_I?

Really need help, thank you so so much!

3. ## Re: Matrices!

I see no reason not to just do this by straightforward calculation.

Since the problem asks about $A^2$, the obvious first thing to do is to square A!
$\begin{bmatrix}3 & 2 \\ -2 & -1\end{bmatrix}\begin{bmatrix}3 & 2 \\ -2 & -1\end{bmatrix}= \begin{bmatrix}5 & 4 \\ -4 & -3 \end{bmatrix}$

I presume you know that I is the identity matrix: $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ so that, for any q, $qI= \begin{bmatrix}q & 0 \\ 0 & q\end{bmatrix}$ and then $A^2- qI= \begin{bmatrix}5- q & 4 \\ -4 & -3- q\end{bmatrix}$ and that must be equal to $pA= \begin{bmatrix}3p & 2p \\ -2p & -p\end{bmatrix}$.

That is, we have the four equations, 5- q= 3p, 4= 2p, -4= -2p, and -3- q= -p. Solve those equations for p and q.