2x2 determinants and inverse matricesAQA A-Level Further Maths: Subtopic test

1. 1
   The matrix $\mathbf{A}=\begin{pmatrix}3&-2\\5&4\end{pmatrix}$.

   (a)

   Find $\det\mathbf{A}$.

   [1 mark]

   • A$22$
   • B$2$
   • C$-22$
   • D$12$

   (b)

   Which of the following is $\mathbf{A}^{-1}$?

   [1 mark]

   • A$\frac{1}{22}\begin{pmatrix}3&2\\-5&4\end{pmatrix}$
   • B$\frac{1}{22}\begin{pmatrix}4&-2\\5&3\end{pmatrix}$
   • C$22\begin{pmatrix}4&2\\-5&3\end{pmatrix}$
   • D$\frac{1}{22}\begin{pmatrix}4&2\\-5&3\end{pmatrix}$

   (c)

   The point $P$ is mapped to the point $(4,14)$ by the transformation with matrix $\mathbf{A}$. Use the inverse of $\mathbf{A}$ to find the coordinates of $P$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{B}=\begin{pmatrix}k&6\\2&k+1\end{pmatrix}$, where $k$ is a constant.

   (a)

   Find an expression for $\det\mathbf{B}$.

   [1 mark]

   • A$k^2+k+12$
   • B$k^2+k-12$
   • C$k^2-12$
   • D$k^2+k-8$

   (b)

   For which values of $k$ is $\mathbf{B}$ singular?

   [1 mark]

   • A$k=-3$ or $k=4$
   • B$k=3$ or $k=4$
   • C$k=3$ or $k=-4$
   • D$k=-3$ or $k=-4$

   (c)

   Given that $k=2$, find $\mathbf{B}^{-1}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrices $\mathbf{P}=\begin{pmatrix}2&1\\3&2\end{pmatrix}$ and $\mathbf{Q}=\begin{pmatrix}1&-1\\0&2\end{pmatrix}$.

   (a)

   Show that $\mathbf{P}$ is non-singular and find $\mathbf{P}^{-1}$.

   [3 marks]

   (b)

   Find $(\mathbf{P}\mathbf{Q})^{-1}$, using $\mathbf{Q}^{-1}$ and your answer to (a).

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{M}=\begin{pmatrix}a&2\\-3&a-5\end{pmatrix}$, where $a$ is a constant.

   (a)

   (i) Find $\det\mathbf{M}$ in terms of $a$.
   (ii) Hence find the values of $a$ for which $\mathbf{M}$ is singular.
   (iii) Find $\mathbf{M}^{-1}$ when $a=4$.

   [6 marks]

   (b)

   Let $a=4$ and $\mathbf{N}=\begin{pmatrix}2&6\\-1&0\end{pmatrix}$.
   (i) Find the matrix $\mathbf{X}$ such that $\mathbf{M}\mathbf{X}=\mathbf{N}$.
   (ii) Find the matrix $\mathbf{Y}$ such that $\mathbf{Y}\mathbf{M}=\mathbf{N}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions