WAEC 2019 · Paper 2 · Q11

(a)
Two linear transformations $P$ and $Q$ in the $Oxy$ plane are defined by $P : (x, y) \to (x + 2y, x)$ and $Q : (x, y) \to (x + y, -x + 2y)$ . Write down the matrices of $P$ and $Q$ .
Show the answer
$P = \begin{pmatrix} 1 & 2 \\ 1 & 0 \end{pmatrix}$ , $Q = \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}$
(b)
Find $M$ such that $2P + 3Q - MQ = 5I$ , where $M$ is a $2 \times 2$ matrix and $I$ is the $2 \times 2$ identity matrix.
Show the answer
$M = \begin{pmatrix} \frac73 & \frac73 \\ -\frac13 & \frac23 \end{pmatrix}$

Worked solution (try it first)

$P = \begin{pmatrix} 1 & 2 \\ 1 & 0 \end{pmatrix}$ and $Q = \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}$ .

Rearrange: $MQ = 2P + 3Q - 5I$
$= \begin{pmatrix} 5 & 7 \\ -1 & 6 \end{pmatrix} - \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}$
$= \begin{pmatrix} 0 & 7 \\ -1 & 1 \end{pmatrix}$ .
$|Q| = 2 + 1 = 3$ , so $Q^{-1} = \frac13\begin{pmatrix} 2 & -1 \\ 1 & 1 \end{pmatrix}$ .
Multiply on the right by $Q^{-1}$ : $M = \begin{pmatrix} 0 & 7 \\ -1 & 1 \end{pmatrix}Q^{-1}$
$= \frac13\begin{pmatrix} 7 & 7 \\ -1 & 2 \end{pmatrix}$ .