Determinants and inverses of 2x2 matricesEdexcel International A Level Further Maths: Subtopic test

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

   (a)

   Find $\det\mathbf{A}$.

   [1 mark]

   • A$2$
   • B$14$
   • C$-2$
   • D$6$

   (b)

   Find $\mathbf{A}^{-1}$.

   [1 mark]

   • A$\frac12\begin{pmatrix}4&3\\ 2&2\end{pmatrix}$
   • B$\frac12\begin{pmatrix}2&3\\ 2&4\end{pmatrix}$
   • C$\frac12\begin{pmatrix}2&-3\\ -2&4\end{pmatrix}$
   • D$2\begin{pmatrix}2&-3\\ -2&4\end{pmatrix}$

   (c)

   Given that $\mathbf{AX}=\begin{pmatrix}2&0\\ 1&1\end{pmatrix}$, find the matrix $\mathbf{X}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $\mathbf{B}=\begin{pmatrix}k&4\\ 3&k+1\end{pmatrix}$, 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$k^2-k-12$
   • D$k^2+k-7$

   (b)

   Find the values of $k$ for which $\mathbf{B}$ is 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
   $\mathbf{C}=\begin{pmatrix}2&1\\ 5&3\end{pmatrix}$ and $\mathbf{D}=\begin{pmatrix}1&2\\ 1&3\end{pmatrix}$.

   (a)

   Find $\mathbf{CD}$ and hence find $(\mathbf{CD})^{-1}$.

   [3 marks]

   (b)

   Find $\mathbf{C}^{-1}$ and $\mathbf{D}^{-1}$, and hence verify that $(\mathbf{CD})^{-1}=\mathbf{D}^{-1}\mathbf{C}^{-1}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $\mathbf{M}=\begin{pmatrix}a+1&3\\ 2&a\end{pmatrix}$, where $a$ is a real 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 $a=0$.

   [6 marks]

   (b)

   Given that $a=1$ and $\mathbf{L}=\begin{pmatrix}1&1\\ 1&2\end{pmatrix}$, find $\mathbf{M}^{-1}$ and $\mathbf{L}^{-1}$, and hence find $(\mathbf{ML})^{-1}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions