WAEC Further Mathematics 2013 Paper

Concept:The product of two matrices $P$ and $Q$ is equal to matrix $R$. Equate the corresponding entries of $PQ$ and $R$ to find the unknown $x$.Explanation:Compute the matrix product $PQ$.$PQ = \begin{pmatrix} 3 & 4 \\ 2 & x \end{pmatrix} \begin{pmatrix} 1 & 3 \\ -2 & 4 \end{pmatrix}$Multiply rows of $P$ by columns of $Q$.Entry $(1,1)$: $3(1) + 4(-2) = 3 - 8 = -5$Entry $(1,2)$: $3(3) + 4(4) = 9 + 16 = 25$Entry $(2,1)$: $2(1) + x(-2) = 2 - 2x$Entry $(2,2)$: $2(3) + x(4) = 6 + 4x$Therefore, $PQ = \begin{pmatrix} -5 & 25 \\ 2 - 2x & 6 + 4x \end{pmatrix}$.Given $PQ = R = \begin{pmatrix} -5 & 25 \\ -8 & 26 \end{pmatrix}$.Equate entry $(2,1)$: $2 - 2x = -8$So, $-2x = -10$, which gives $x = 5$.Verify with entry $(2,2)$: $6 + 4x = 26 \Rightarrow 4x = 20 \Rightarrow x = 5$.Both entries confirm the same value.Answer:$x = 5$, which is option D.