JAMB 1998 · UME · Q24

Matrices & determinants

Let $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ , $P = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$ and $Q = \begin{pmatrix} u & 4 + u \\ -2v & v \end{pmatrix}$ be $2 \times 2$ matrices such that $PQ = I$ . Find $(u, v)$ .

Worked solution (try it first)

Row 1 of $P$ times column 1 of $Q$ must give the 1 in $I$ : $2u + 3(-2v) = 1$ , so $2u - 6v = 1$ .
Row 2 of $P$ times column 1 of $Q$ must give 0: $4u - 10v = 0$ , so $u = \frac52 v$ .
Substitute: $5v - 6v = 1$ , so $v = -1$ and $u = -\frac52$ .
So $(u, v) = \left(-\frac52, -1\right)$ , option A.

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