Matrix arithmeticAQA A-Level Further Maths: Subtopic test

1. 1
   $\mathbf{C}=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $\mathbf{D}=\begin{pmatrix}0&-1\\5&2\end{pmatrix}$.

   (a)

   Find $\mathbf{CD}$.

   [1 mark]

   • A$\begin{pmatrix}-3&-4\\11&18\end{pmatrix}$
   • B$\begin{pmatrix}10&3\\20&5\end{pmatrix}$
   • C$\begin{pmatrix}0&-2\\15&8\end{pmatrix}$
   • D$\begin{pmatrix}10&5\\20&11\end{pmatrix}$

   (b)

   Find $2\mathbf{C}-\mathbf{D}$.

   [1 mark]

   • A$\begin{pmatrix}2&5\\1&6\end{pmatrix}$
   • B$\begin{pmatrix}2&3\\11&10\end{pmatrix}$
   • C$\begin{pmatrix}1&4\\-7&0\end{pmatrix}$
   • D$\begin{pmatrix}2&6\\-4&4\end{pmatrix}$

   (c)

   Given that $\mathbf{C}+\mathbf{X}=3\mathbf{I}$, where $\mathbf{I}$ is the $2\times2$ identity matrix, find $\mathbf{X}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $\mathbf{P}=\begin{pmatrix}1&0&2\\-1&3&1\end{pmatrix}$ and $\mathbf{Q}=\begin{pmatrix}2&1\\0&-1\\4&3\end{pmatrix}$.

   (a)

   What is the order of $\mathbf{PQ}$?

   [1 mark]

   • A$3\times3$
   • B$2\times3$
   • C$3\times2$
   • D$2\times2$

   (b)

   Which of these is not defined?

   [1 mark]

   • A$\mathbf{PQ}$
   • B$\mathbf{QP}$
   • C$\mathbf{P}+\mathbf{Q}$
   • D$2\mathbf{P}$

   (c)

   Find $\mathbf{PQ}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $\mathbf{M}=\begin{pmatrix}2&-1\\1&0\end{pmatrix}$.

   (a)

   Find $\mathbf{M}^2$ and show that $\mathbf{M}^2=2\mathbf{M}-\mathbf{I}$, where $\mathbf{I}$ is the $2\times2$ identity matrix.

   [3 marks]

   (b)

   Use the result of part (a) to show that $\mathbf{M}^3=3\mathbf{M}-2\mathbf{I}$, and hence find $\mathbf{M}^3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $\mathbf{A}=\begin{pmatrix}2&-1\\3&4\end{pmatrix}$ and $\mathbf{B}=\begin{pmatrix}1&0\\-2&3\end{pmatrix}$.

   (a)

   (i) Find $\mathbf{AB}$ and $\mathbf{BA}$.
   (ii) Find $\mathbf{AB}-\mathbf{BA}$, and explain what this shows about matrix multiplication.

   [6 marks]

   (b)

   Find the constants $p$ and $q$ such that $\mathbf{A}^2+p\mathbf{A}+q\mathbf{I}=\mathbf{O}$, where $\mathbf{I}$ is the $2\times2$ identity matrix and $\mathbf{O}$ is the $2\times2$ zero matrix.

   [6 marks]

   Total for question 4: 12 marks

End of questions