3x3 determinants and inverse matricesAQA A-Level Further Maths: Subtopic test

1. 1
   The matrix $\mathbf{A}=\begin{pmatrix}1&2&0\\3&1&1\\0&2&1\end{pmatrix}$.

   (a)

   Find $\det\mathbf{A}$.

   [1 mark]

   • A$7$
   • B$-7$
   • C$5$
   • D$-1$

   (b)

   A solid of volume $4\text{ cm}^3$ is transformed by the matrix $\mathbf{A}$. What is the volume of the image?

   [1 mark]

   • A$-28\text{ cm}^3$
   • B$11\text{ cm}^3$
   • C$28\text{ cm}^3$
   • D$\frac47\text{ cm}^3$

   (c)

   State whether $\mathbf{A}$ preserves or reverses orientation, giving a reason.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{B}=\begin{pmatrix}2&0&1\\1&1&0\\0&3&1\end{pmatrix}$, for which $\det\mathbf{B}=5$.

   (a)

   Find the cofactor of the entry in row 2, column 1 of $\mathbf{B}$.

   [1 mark]

   • A$-3$
   • B$2$
   • C$-6$
   • D$3$

   (b)

   $\mathbf{B}$ is non-singular. Find the entry in row 1, column 1 of $\mathbf{B}^{-1}$.

   [1 mark]

   • A$\frac15$
   • B$1$
   • C$5$
   • D$-\frac15$

   (c)

   Find the entry in row 3, column 2 of $\mathbf{B}^{-1}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{C}=\begin{pmatrix}1&1&1\\1&2&3\\1&4&9\end{pmatrix}$.

   (a)

   Show that $\det\mathbf{C}=2$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{D}=\begin{pmatrix}k&1&0\\0&2&1\\1&0&k\end{pmatrix}$, where $k$ is a real constant, represents a transformation of three-dimensional space.

   (a)

   (i) Find $\det\mathbf{D}$ in terms of $k$.
   (ii) Show that $\mathbf{D}$ is non-singular for every real value of $k$.
   (iii) When $k=2$, a solid of volume $12\text{ cm}^3$ is transformed by $\mathbf{D}$. Find the volume of the image and state whether orientation is preserved.

   [6 marks]

   (b)

   (i) A solid of volume $6\text{ cm}^3$ is transformed by $\mathbf{D}$ to a solid of volume $114\text{ cm}^3$. Find the possible values of $k$.
   (ii) Find $\mathbf{D}^{-1}$ when $k=1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions