Determinants and inverse matricesEdexcel A-Level Further Maths: Subtopic test

1. 1
   The matrix $\mathbf{A}=\begin{pmatrix}4&3\\ 2&5\end{pmatrix}$ represents a linear transformation of the plane.

   (a)

   Find $\det\mathbf{A}$.

   [1 mark]

   • A$26$
   • B$20$
   • C$-14$
   • D$14$

   (b)

   What is the top-right entry of $\mathbf{A}^{-1}$?

   [1 mark]

   • A$-\frac{3}{14}$
   • B$\frac{3}{14}$
   • C$-\frac{1}{7}$
   • D$\frac{5}{14}$

   (c)

   A triangle of area $5$ cm$^2$ is transformed by $\mathbf{A}$. Find the area of its image, and state whether the orientation of the triangle is preserved.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{B}=\begin{pmatrix}k&3\\ 4&k+1\end{pmatrix}$, where $k$ is a constant.

   (a)

   Find $\det\mathbf{B}$ in terms of $k$.

   [1 mark]

   • A$k^2+k+12$
   • B$k^2-12$
   • C$k^2+k-12$
   • D$2k-11$

   (b)

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

   [1 mark]

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

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{C}=\begin{pmatrix}2&0&1\\ 1&3&-1\\ 0&2&4\end{pmatrix}$ represents a linear transformation of three-dimensional space.

   (a)

   Find $\det\mathbf{C}$.

   [3 marks]

   (b)

   A solid of volume $8$ cm$^3$ is transformed by $\mathbf{C}$. Find the volume of the image. Write down $\det\mathbf{C}^{-1}$ and state, with a reason, whether $\mathbf{C}$ preserves orientation.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{P}=\begin{pmatrix}1&2&-1\\ 0&1&3\\ 2&-1&1\end{pmatrix}$ represents a linear transformation of three-dimensional space.

   (a)

   Find $\det\mathbf{P}$, and hence state with a reason whether $\mathbf{P}$ is singular. Find the matrix of cofactors of $\mathbf{P}$.

   [6 marks]

   (b)

   Hence find $\mathbf{P}^{-1}$. A point is transformed by $\mathbf{P}$ to the point $(2,11,3)$. Find the original point, and write down $\det\mathbf{P}^{-1}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions