Let $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 4 \\ 2 & -1 \end{pmatrix}$. Find the matrix $AB$.
Question HKC78-AISL-MAT-01
[Maximum mark: 4]
Let $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 4 \\ 2 & -1 \end{pmatrix}$. Find the matrix $AB$.
Mark Scheme
Row-by-column products: top-left $2cdot 1 + 1cdot 2 = 4$, top-right $2cdot 4 + 1cdot(-1) = 7$. [2]
Bottom-left $0cdot 1 + 3cdot 2 = 6$, bottom-right $0cdot 4 + 3cdot(-1) = -3$. [1]
$AB = begin{pmatrix} 4 & 7 \ 6 & -3 end{pmatrix}$. [1]
Total: 4 marks
Let $A = \begin{pmatrix} 4 & 1 \\ 2 & 1 \end{pmatrix}$.
(a) Find $A^{-1}$.
(b) Hence solve $4x + y = 9$ and $2x + y = 5$.
Question HKC78-AISL-MAT-02
[Maximum mark: 6]
Let $A = \begin{pmatrix} 4 & 1 \\ 2 & 1 \end{pmatrix}$.
(a) Find $A^{-1}$.
(b) Hence solve $4x + y = 9$ and $2x + y = 5$.
Mark Scheme
(a) $det A = 4 – 2 = 2$. $A^{-1} = dfrac{1}{2}begin{pmatrix} 1 & -1 \ -2 & 4 end{pmatrix}$. [3]
(b) $begin{pmatrix} x \ y end{pmatrix} = dfrac{1}{2}begin{pmatrix} 1 & -1 \ -2 & 4 end{pmatrix}begin{pmatrix} 9 \ 5 end{pmatrix} = dfrac{1}{2}begin{pmatrix} 4 \ 2 end{pmatrix} = begin{pmatrix} 2 \ 1 end{pmatrix}$. [2]
So $x = 2$ and $y = 1$. Check: $8 + 1 = 9$ and $4 + 1 = 5$. [1]
Total: 6 marks
