Matrix arithmeticEdexcel International A Level Further Maths: Subtopic test

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

   (a)

   Find $\mathbf{A}+\mathbf{B}$.

   [1 mark]

   • A$\begin{pmatrix}1&-1\\ 5&-1\end{pmatrix}$
   • B$\begin{pmatrix}3&-1\\ 1&9\end{pmatrix}$
   • C$\begin{pmatrix}3&1\\ 1&9\end{pmatrix}$
   • D$\begin{pmatrix}2&0\\ -6&20\end{pmatrix}$

   (b)

   Find $3\mathbf{A}-\mathbf{B}$.

   [1 mark]

   • A$\begin{pmatrix}7&-3\\ 7&17\end{pmatrix}$
   • B$\begin{pmatrix}-1&-1\\ 9&-11\end{pmatrix}$
   • C$\begin{pmatrix}5&-3\\ 11&7\end{pmatrix}$
   • D$\begin{pmatrix}6&-3\\ 9&12\end{pmatrix}$

   (c)

   Find $\mathbf{AB}$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Find $\mathbf{MN}$.

   [1 mark]

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

   (b)

   Find $\mathbf{M}^2$.

   [1 mark]

   • A$\begin{pmatrix}1&4\\ 0&1\end{pmatrix}$
   • B$\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$
   • C$\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$
   • D$\begin{pmatrix}1&4\\ 0&-1\end{pmatrix}$

   (c)

   Find $\mathbf{MN}-\mathbf{NM}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $\mathbf{P}=\begin{pmatrix}a&2\\ 1&-1\end{pmatrix}$ and $\mathbf{Q}=\begin{pmatrix}3&b\\ 0&2\end{pmatrix}$, where $a$ and $b$ are constants.

   (a)

   Given that $\mathbf{P}+2\mathbf{Q}=\begin{pmatrix}10&8\\ 1&3\end{pmatrix}$, find the values of $a$ and $b$.

   [3 marks]

   (b)

   Using these values of $a$ and $b$, find $\mathbf{PQ}$ and $\mathbf{QP}$, and state, with a reason, whether $\mathbf{PQ}=\mathbf{QP}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A café sells tea and coffee at two branches. The matrix $\mathbf{S}=\begin{pmatrix}120&80\\ 90&150\end{pmatrix}$ shows the numbers of drinks sold in one week: the rows are branch 1 and branch 2, and the columns are tea and coffee. The price of a tea is £2 and the price of a coffee is £3.

   (a)

   (i) Find $\mathbf{S}\begin{pmatrix}2\\3\end{pmatrix}$ and state what each entry represents.
   (ii) Sales are predicted to rise by 10% at both branches next week. Find a matrix, in terms of $\mathbf{S}$ and the prices, whose entries give the predicted takings, and evaluate it.

   [6 marks]

   (b)

   (i) Find a $1\times2$ matrix $\mathbf{J}$ such that $\mathbf{J}\mathbf{S}\begin{pmatrix}2\\3\end{pmatrix}$ gives the total takings for the two branches together, and evaluate this product.
   (ii) The price of each drink then rises by 50p. Using matrices, find the new weekly takings at each branch and the new total, assuming the numbers sold do not change.

   [6 marks]

   Total for question 4: 12 marks

End of questions