MatricesIB MYP Maths Extended: Subtopic test

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

   (a)

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

   [1 mark]

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

   (b)

   Find $\mathbf{AB}$.

   [1 mark]

   • A$\begin{pmatrix}4&3\\ -6&15\end{pmatrix}$
   • B$\begin{pmatrix}2&11\\ -4&17\end{pmatrix}$
   • C$\begin{pmatrix}2&-4\\ 0&15\end{pmatrix}$
   • D$\begin{pmatrix}0&13\\ -6&15\end{pmatrix}$

   (c)

   Find $\mathbf A^2$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   What is the order of the matrix $\mathbf{MN}$?

   [1 mark]

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

   (b)

   What is the entry in row 2, column 1 of $\mathbf{MN}$?

   [1 mark]

   • A$8$
   • B$9$
   • C$5$
   • D$3$

   (c)

   Find the matrix $\mathbf{MN}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Four towns $W$, $X$, $Y$ and $Z$ are joined by two-way bus routes. The only direct routes are $W$ to $X$, $W$ to $Y$, $X$ to $Y$, and $Y$ to $Z$.

   (a)

   Write down the adjacency matrix $\mathbf M$ for the network, using the order $W$, $X$, $Y$, $Z$ for the rows and columns, and state its order.

   [3 marks]

   (b)

   The entries of $\mathbf M^2$ give the number of journeys that use exactly two bus routes. Show how to find the number of journeys from $W$ to $Z$, and the number from $Y$ back to $Y$, that use exactly two routes. List the journeys.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bakery has branches in Dubai and Sharjah. The matrix $\mathbf S=\begin{pmatrix}120&80&40\\ 90&100&60\end{pmatrix}$ shows the number of rolls, tarts and cakes sold in one week. The first row is Dubai and the second row is Sharjah, and the columns are rolls, tarts and cakes. The prices, in AED, are given by $\mathbf P=\begin{pmatrix}2\\ 5\\ 12\end{pmatrix}$, in the same order of products.

   (a)

   (i) State the order of $\mathbf S$, the order of $\mathbf P$, and the order of $\mathbf{SP}$.
   (ii) Find $\mathbf{SP}$.
   (iii) Explain what the entries of $\mathbf{SP}$ represent, and find the total weekly takings of the two branches.

   [6 marks]

   (b)

   In Ramadan, the manager predicts that every product will sell 20% more at both branches, and that the price of a cake will rise to 14 AED, with the other prices unchanged. Use matrices to find the predicted weekly takings of each branch, and the percentage increase in the total takings.
   Evaluate one limitation of the manager's prediction.

   [6 marks]

   Total for question 4: 12 marks

End of questions