1. MAtrix Problem

If $P=\begin{bmatrix} 1 & -1 \\2 & -1 \end{bmatrix}$, $Q=\begin{bmatrix} 1 & 1 \\ q & -1 \end{bmatrix}$

$(P+Q)^2 = P^2 + Q^2$ , determine the value of $q$

2. Hi

$P+Q=\begin{bmatrix} 2 & 0 \\2+q & -2 \end{bmatrix}$
$(P+Q)^2=\begin{bmatrix} 4 & 0 \\0 & 4 \end{bmatrix}$

$P^2=\begin{bmatrix} -1 & 0 \\0 & -1 \end{bmatrix}$

$Q^2=\begin{bmatrix} 1+q & 0 \\0 & 1+q \end{bmatrix}$
$P^2 + Q^2=\begin{bmatrix} q & 0 \\0 & q \end{bmatrix}$

$(P+Q)^2 = P^2 + Q^2$ for q=4

3. Hi,

A slightly different method which might save you some calculations :

$(P+Q)^2=P^2+PQ+QP+Q^2$ hence $(P+Q)^2=P^2+Q^2 \Longleftrightarrow PQ+QP=0\quad (1)$. Now all you have to do is to compute $PQ+QP$ and solve (1) for $q$.