1.14 MatricesIB Maths: Applications and Interpretation HL: Subtopic test

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

   (a)

   Find $AB$.

   [1 mark]

   • A$\begin{pmatrix}2&0\\ -3&8\end{pmatrix}$
   • B$\begin{pmatrix}2&1\\ 4&7\end{pmatrix}$
   • C$\begin{pmatrix}1&2\\ -1&8\end{pmatrix}$
   • D$\begin{pmatrix}3&1\\ 2&6\end{pmatrix}$

   (b)

   Find $\det A$.

   [1 mark]

   • A$5$
   • B$11$
   • C$-5$
   • D$\frac{1}{5}$

   (c)

   Find $BA$ and hence state, with a reason, whether $AB=BA$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cafe sells small cups of coffee at $s$ EUR each and large cups at $l$ EUR each. On Monday, 3 small and 2 large cups cost 13.90 EUR in total. On Tuesday, 2 small and 5 large cups cost 21.00 EUR in total.

   (a)

   Which matrix equation represents this information?

   [1 mark]

   • A$\begin{pmatrix}3&2\\ 2&5\end{pmatrix}\begin{pmatrix}s\\ l\end{pmatrix}=\begin{pmatrix}13.9\\ 21\end{pmatrix}$
   • B$\begin{pmatrix}3&2\\ 5&2\end{pmatrix}\begin{pmatrix}s\\ l\end{pmatrix}=\begin{pmatrix}13.9\\ 21\end{pmatrix}$
   • C$\begin{pmatrix}3&2\\ 2&5\end{pmatrix}\begin{pmatrix}s\\ l\end{pmatrix}=\begin{pmatrix}21\\ 13.9\end{pmatrix}$
   • D$\begin{pmatrix}3&2\\ 2&5\end{pmatrix}\begin{pmatrix}l\\ s\end{pmatrix}=\begin{pmatrix}13.9\\ 21\end{pmatrix}$

   (b)

   Find the determinant of the coefficient matrix $\begin{pmatrix}3&2\\ 2&5\end{pmatrix}$.

   [1 mark]

   • A$19$
   • B$11$
   • C$-11$
   • D$\frac{1}{11}$

   (c)

   Use $\mathbf{x}=A^{-1}\mathbf{b}$ with your GDC, where $A$ is the coefficient matrix, to find the price of a small cup and of a large cup.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Letters are replaced by numbers ($A=1$, $B=2$, ..., $Z=26$) and a message is coded in pairs of letters. Each pair of numbers forms a column vector, which is multiplied on the left by the matrix $K=\begin{pmatrix}2&3\\ 1&2\end{pmatrix}$ to give the coded pair.

   (a)

   Show that $\det K=1$ and find $K^{-1}$.

   [3 marks]

   (b)

   A message is coded as the matrix $C=\begin{pmatrix}41&62\\ 23&41\end{pmatrix}$, where each column is one coded pair. Find the original message.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school shop sells pencils, pens and notebooks. Pencils cost $p$ EUR each, pens cost $q$ EUR each and notebooks cost $r$ EUR each. Class X buys 10 pencils, 4 pens and 2 notebooks for 14.60 EUR. Class Y buys 6 pencils, 5 pens and 3 notebooks for 16.20 EUR. Class Z buys 4 pencils, 2 pens and 6 notebooks for 18.80 EUR.

   (a)

   (i) Write down a matrix equation of the form $A\mathbf{x}=\mathbf{b}$ for $\mathbf{x}=\begin{pmatrix}p\\ q\\ r\end{pmatrix}$.
   (ii) Use your GDC to find $p$, $q$ and $r$.

   [6 marks]

   (b)

   Let $A=\begin{pmatrix}10&4&2\\ 6&5&3\\ 4&2&6\end{pmatrix}$ be the numbers of items bought by the three classes, and $\mathbf{b}=\begin{pmatrix}14.6\\ 16.2\\ 18.8\end{pmatrix}$ the amounts paid in EUR. The shop pays its supplier 0.30 EUR for each pencil, 0.70 EUR for each pen and 1.50 EUR for each notebook, written as the column vector $\mathbf{c}$.
   (i) Calculate $A\mathbf{c}$ and hence the profit on each class's purchase.
   (ii) Find the shop's total profit on the three purchases.
   (iii) Explain why $\mathbf{c}A$ cannot be calculated.

   [6 marks]

   Total for question 4: 12 marks

End of questions