FP1: Matrix algebraEdexcel International A Level Further Maths: Topic test

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

   (a)

   Find $\mathbf{A}-2\mathbf{B}$.

   [1 mark]

   • A$\begin{pmatrix}3&8\\ -1&8\end{pmatrix}$
   • B$\begin{pmatrix}3&-12\\ 3&0\end{pmatrix}$
   • C$\begin{pmatrix}3&-7\\ 2&2\end{pmatrix}$
   • D$\begin{pmatrix}6&-9\\ 3&6\end{pmatrix}$

   (b)

   Find $\det\mathbf{A}$.

   [1 mark]

   • A$10$
   • B$12$
   • C$14$
   • D$-14$

   (c)

   Find $\mathbf{AB}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cinema records ticket sales in the matrix $\mathbf{T}=\begin{pmatrix}120&80\\ 150&95\end{pmatrix}$. The rows are Saturday and Sunday and the columns are adult and child tickets. An adult ticket costs £9 and a child ticket costs £5, and these prices are the entries of $\mathbf{p}=\begin{pmatrix}9\\ 5\end{pmatrix}$.

   (a)

   Find $\mathbf{Tp}$, the total takings in pounds on Saturday and on Sunday.

   [1 mark]

   • A$\begin{pmatrix}1480\\ 1825\end{pmatrix}$
   • B$\begin{pmatrix}1320\\ 1605\end{pmatrix}$
   • C$\begin{pmatrix}200\\ 245\end{pmatrix}$
   • D$\begin{pmatrix}1080&400\\ 1350&475\end{pmatrix}$

   (b)

   Sales fall by $10\%$ in every category the following weekend. Which matrix shows the new sales?

   [1 mark]

   • A$\begin{pmatrix}12&8\\ 15&9.5\end{pmatrix}$
   • B$\begin{pmatrix}132&88\\ 165&104.5\end{pmatrix}$
   • C$\begin{pmatrix}119.9&79.9\\ 149.9&94.9\end{pmatrix}$
   • D$\begin{pmatrix}108&72\\ 135&85.5\end{pmatrix}$

   (c)

   The prices rise to £10 for an adult ticket and £6 for a child ticket. Use a matrix product to find the total takings on each of the two days.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{M}=\begin{pmatrix}k&2\\ 3&k-1\end{pmatrix}$, where $k$ is a real constant.

   (a)

   Given that $\mathbf{M}$ is singular, find the possible values of $k$.

   [3 marks]

   (b)

   Given that $k=4$, find $\mathbf{M}^{-1}$ and use it to solve $\mathbf{M}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}10\\ 9\end{pmatrix}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $\mathbf{P}=\begin{pmatrix}1&2\\ 3&5\end{pmatrix}$ and $\mathbf{Q}=\begin{pmatrix}2&1\\ 1&1\end{pmatrix}$.

   (a)

   Find $\mathbf{PQ}$ and $(\mathbf{PQ})^{-1}$. Verify, by evaluating $\mathbf{Q}^{-1}\mathbf{P}^{-1}$, that $(\mathbf{PQ})^{-1}=\mathbf{Q}^{-1}\mathbf{P}^{-1}$.

   [6 marks]

   (b)

   (i) Show that $\mathbf{P}^2-6\mathbf{P}=\mathbf{I}$, where $\mathbf{I}$ is the $2\times2$ identity matrix. (ii) Hence express $\mathbf{P}^{-1}$ in terms of $\mathbf{P}$ and $\mathbf{I}$. (iii) Find the matrix $\mathbf{X}$ such that $\mathbf{PX}=\mathbf{Q}$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   $\mathbf{R}=\begin{pmatrix}5&-3\\ 2&-1\end{pmatrix}$.

   (a)

   Find $\mathbf{R}^2$.

   [1 mark]

   • A$\begin{pmatrix}25&9\\ 4&1\end{pmatrix}$
   • B$\begin{pmatrix}10&-6\\ 4&-2\end{pmatrix}$
   • C$\begin{pmatrix}19&-12\\ 8&5\end{pmatrix}$
   • D$\begin{pmatrix}19&-12\\ 8&-5\end{pmatrix}$

   (b)

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

   [1 mark]

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

   (c)

   Given that $\mathbf{R}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}1\\ 2\end{pmatrix}$, use $\mathbf{R}^{-1}$ to find $x$ and $y$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A rental company records the number of bikes (first column) and scooters (second column) at depot X (first row) and depot Y (second row) at 8 am in $\mathbf{N}=\begin{pmatrix}12&7\\ 9&14\end{pmatrix}$. During the day the changes in stock are given by $\mathbf{C}=\begin{pmatrix}-3&2\\ 4&-5\end{pmatrix}$.

   (a)

   Find the matrix showing the stock at the end of the day.

   [1 mark]

   • A$\begin{pmatrix}15&5\\ 5&19\end{pmatrix}$
   • B$\begin{pmatrix}-8&-11\\ 29&-52\end{pmatrix}$
   • C$\begin{pmatrix}9&9\\ 13&9\end{pmatrix}$
   • D$\begin{pmatrix}-15&-5\\ -5&-19\end{pmatrix}$

   (b)

   A bike earns £8 per day and a scooter earns £15 per day. Which matrix gives the daily earnings of each depot at 8 am?

   [1 mark]

   • A$\begin{pmatrix}236\\ 247\end{pmatrix}$
   • B$\begin{pmatrix}201\\ 282\end{pmatrix}$
   • C$\begin{pmatrix}19\\ 23\end{pmatrix}$
   • D$\begin{pmatrix}96&105\\ 72&210\end{pmatrix}$

   (c)

   Find the value of $k$ for which $\mathbf{N}+k\mathbf{C}$ has top-left entry $6$, and write down this matrix.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   $\mathbf{H}=\begin{pmatrix}2&3\\ 1&2\end{pmatrix}$, and the matrix $\mathbf{K}$ satisfies $\mathbf{HK}=\begin{pmatrix}7&4\\ 4&3\end{pmatrix}$.

   (a)

   Find $\mathbf{H}^{-1}$, showing your value of $\det\mathbf{H}$.

   [3 marks]

   (b)

   Find $\mathbf{K}$, and hence find $\mathbf{K}^{-1}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   $\mathbf{L}=\begin{pmatrix}2&-1\\ -3&2\end{pmatrix}$.

   (a)

   (i) Show that $\mathbf{L}^2-4\mathbf{L}+\mathbf{I}=\mathbf{0}$. (ii) Hence find $\mathbf{L}^{-1}$ without using the determinant formula.

   [6 marks]

   (b)

   The matrix $\mathbf{T}=\begin{pmatrix}1&k\\ 2&3\end{pmatrix}$, where $k$ is a constant. (i) Find $\mathbf{LT}$ in terms of $k$. (ii) Find the value of $k$ for which $\mathbf{LT}$ is singular. (iii) For $k=1$, find $(\mathbf{LT})^{-1}$.

   [6 marks]

   Total for question 8: 12 marks

End of questions