We are given ,
Px + qy = p – q    ....(1)
qx – py = p + q    ....(2)

Now, we can generate the matrix.

$\left[\begin{array}{cc}\mathrm{p}& \mathrm{q}\\ \mathrm{q}& -\mathrm{p}\end{array}\right]\left[\begin{array}{l}\mathrm{x}\\ \mathrm{y}\end{array}\right]=\left[\begin{array}{c}\mathrm{p}-\mathrm{q}\\ \mathrm{p}+\mathrm{q}\end{array}\right]$

let $\mathrm{A}=\left[\begin{array}{cc}\mathrm{p}& \mathrm{q}\\ \mathrm{q}& -\mathrm{p}\end{array}\right]$

So, $\left[\begin{array}{l}\mathrm{x}\\ \mathrm{y}\end{array}\right]={\mathrm{A}}^{-1}\left[\begin{array}{l}\mathrm{p}-\mathrm{q}\\ \mathrm{p}+\mathrm{q}\end{array}\right]$

where ${\mathrm{A}}^{-1}=\frac{1}{\left|\mathrm{A}\right|}\left[\begin{array}{cc}-\mathrm{p}& -\mathrm{q}\\ -\mathrm{q}& \mathrm{p}\end{array}\right]$

& |A| = (p) (– p) – (q) (q) = – p² – q²

So, $\left[\begin{array}{l}\mathrm{x}\\ \mathrm{y}\end{array}\right]=\frac{1}{-{\mathrm{p}}^2-{\mathrm{q}}^2}\left[\begin{array}{cc}-\mathrm{p}& -\mathrm{q}\\ -\mathrm{q}& \mathrm{p}\end{array}\right]\left[\begin{array}{l}\mathrm{p}-\mathrm{q}\\ \mathrm{p}+\mathrm{q}\end{array}\right]$

$=\frac{1}{-{\mathrm{p}}^2-{\mathrm{q}}^2}\left[\begin{array}{l}-{\mathrm{p}}^2+\mathrm{pq}-\mathrm{pq}-{\mathrm{q}}^2\\ -\mathrm{pq}+{\mathrm{q}}^2+{\mathrm{p}}^2+\mathrm{pq}\end{array}\right]$

$\mathrm{x}=\frac{-{\mathrm{p}}^2-{\mathrm{q}}^2}{-{\mathrm{p}}^2-{\mathrm{q}}^2}=1\mathrm{\&}\mathrm{y}=\frac{{\mathrm{p}}^2+{\mathrm{q}}^2}{-\left({\mathrm{p}}^2+{\mathrm{q}}^2\right)}=-1$

here (x, y) = (1, – 1)