Matrix arithmetic Problem set A

State the order of each matrix.

1. (a) \(\begin{pmatrix}3 & -1 \\ 0 & 4 \\ 5 & 2\end{pmatrix}\)
2. (b) \(\begin{pmatrix}7 & 0 & -2 & 1\end{pmatrix}\)
3. (c) \(\begin{pmatrix}1 & 0 & 4 \\ 2 & 2 & -1 \\ 0 & 6 & 3\end{pmatrix}\)

Find \(M\) in each case.

1. (a) \(M + \begin{pmatrix}4 & -1 \\ 2 & 0\end{pmatrix} = \begin{pmatrix}1 & 5 \\ -3 & 7\end{pmatrix}\)
2. (b) \(2M - \begin{pmatrix}2 & -5 \\ 0 & 3\end{pmatrix} = \begin{pmatrix}-1 & 4 \\ 6 & -2\end{pmatrix}\)

\(A = \begin{pmatrix}2 & -5 \\ 0 & 3\end{pmatrix},\; B = \begin{pmatrix}-1 & 4 \\ 6 & -2\end{pmatrix},\; C = \begin{pmatrix}1 & 0 & 2 \\ -3 & 5 & 1\end{pmatrix}\). Find each of these, or say why it cannot be done.

1. (a) \(A + B\)
2. (b) \(B - A\)
3. (c) \(A + C\)
4. (d) \(3A\)
5. (e) \(2A - 3B\)
6. (f) \(\tfrac{1}{2}B\)

Find \(x\) and \(y\).

1. (a) \(\begin{pmatrix}x+3 & -2 \\ 4 & 2y\end{pmatrix} = \begin{pmatrix}7 & -2 \\ 4 & y+5\end{pmatrix}\)
2. (b) \(\begin{pmatrix}x^2 & 1 \\ y & 6\end{pmatrix} = \begin{pmatrix}16 & 1 \\ -x & 6\end{pmatrix}\)

A shop stocks three sizes of two kinds of trainer. The stock at the start of the week is \(S = \begin{pmatrix}14 & 9 & 20 \\ 8 & 11 & 16\end{pmatrix}\). During the week it sells \(D = \begin{pmatrix}6 & 4 & 11 \\ 3 & 7 & 5\end{pmatrix}\) and takes a delivery of \(R = \begin{pmatrix}10 & 0 & 12 \\ 5 & 10 & 0\end{pmatrix}\). Write a matrix expression for the stock at the end of the week, and work it out.

The monthly rents, in euros, for one-bedroom and two-bedroom flats in two districts are \(\begin{pmatrix}850 & 1200 \\ 640 & 990\end{pmatrix}\). All rents rise by 6%. Find the matrix of new rents.

Two swimming clubs win gold, silver and bronze medals at the autumn gala, \(\begin{pmatrix}4 & 6 & 3 \\ 7 & 2 & 5\end{pmatrix}\), and at the spring gala, \(\begin{pmatrix}5 & 3 & 8 \\ 6 & 6 & 1\end{pmatrix}\). Find the matrix of total medals. Which club won more medals altogether?

Find \(p\) and \(q\) such that \(p\begin{pmatrix}2 & 1 \\ 0 & 3\end{pmatrix} + q\begin{pmatrix}1 & -1 \\ 4 & 0\end{pmatrix} = \begin{pmatrix}7 & 2 \\ 4 & 9\end{pmatrix}\).

\(A = \begin{pmatrix}a & 2 \\ -1 & 3\end{pmatrix}\) and \(B = \begin{pmatrix}1 & b \\ 2 & 4\end{pmatrix}\). Given that \(A + sB = tI\), find \(a, b, s\) and \(t\).