Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$. Find $\det A$ and $A^{-1}$.
Question HKC78-AIHL-MAT-01
[Maximum mark: 5]
Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$. Find $\det A$ and $A^{-1}$.
Mark Scheme
$det A = 4 – 6 = -2$. [1]
$A^{-1} = -dfrac{1}{2}begin{pmatrix} 4 & -2 \ -3 & 1 end{pmatrix} = begin{pmatrix} -2 & 1 \ dfrac{3}{2} & -dfrac{1}{2} end{pmatrix}$. [3]
Check the top row against the first column of $A$: $-2cdot 1 + 1cdot 3 = 1$ and $-2cdot 2 + 1cdot 4 = 0$. [1]
Total: 5 marks
Use an inverse matrix to solve $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 4 \end{pmatrix}$.
Question HKC78-AIHL-MAT-02
[Maximum mark: 5]
Use an inverse matrix to solve $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 4 \end{pmatrix}$.
Mark Scheme
Let $M = begin{pmatrix} 2 & 1 \ 1 & 1 end{pmatrix}$. Then $det M = 1$, so $M^{-1} = begin{pmatrix} 1 & -1 \ -1 & 2 end{pmatrix}$. [2]
$begin{pmatrix} x \ y end{pmatrix} = begin{pmatrix} 1 & -1 \ -1 & 2 end{pmatrix}begin{pmatrix} 7 \ 4 end{pmatrix} = begin{pmatrix} 3 \ 1 end{pmatrix}$. [2]
Check: $2(3) + 1 = 7$ and $3 + 1 = 4$. So $x = 3$ and $y = 1$. [1]
Total: 5 marks
