Matrix arithmetic Problem set B

State the order of each matrix.

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

Find \(M\) in each case.

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

\(A = \begin{pmatrix}3 & -2 \\ 1 & 4\end{pmatrix},\; B = \begin{pmatrix}-3 & 6 \\ 4 & -5\end{pmatrix},\; C = \begin{pmatrix}2 & -1 & 0 \\ 4 & 1 & -3\end{pmatrix}\). 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) \(4A\)
5. (e) \(3A - 2B\)
6. (f) \(\tfrac{1}{2}B\)

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

1. (a) \(\begin{pmatrix}x-2 & 0 \\ 7 & 2y\end{pmatrix} = \begin{pmatrix}5 & 0 \\ 7 & y+6\end{pmatrix}\)
2. (b) \(\begin{pmatrix}x^2 & -1 \\ y & 4\end{pmatrix} = \begin{pmatrix}36 & -1 \\ 2x & 4\end{pmatrix}\)

A bike shop stocks three frame sizes of two kinds of bicycle. The stock at the start of the week is \(S = \begin{pmatrix}12 & 8 & 15 \\ 9 & 14 & 10\end{pmatrix}\). During the week it sells \(D = \begin{pmatrix}5 & 3 & 9 \\ 4 & 6 & 2\end{pmatrix}\) and takes a delivery of \(R = \begin{pmatrix}8 & 0 & 6 \\ 0 & 5 & 10\end{pmatrix}\). Write a matrix expression for the stock at the end of the week, and work it out.

The monthly fees, in dollars, for a single and a family membership at two gyms are \(\begin{pmatrix}45 & 80 \\ 62 & 95\end{pmatrix}\). All fees rise by 8%. Find the matrix of new fees.

Two quiz teams get the following numbers of correct answers in the three rounds of the autumn league, \(\begin{pmatrix}7 & 5 & 9 \\ 4 & 8 & 6\end{pmatrix}\), and of the spring league, \(\begin{pmatrix}6 & 9 & 5 \\ 8 & 3 & 7\end{pmatrix}\). Find the matrix of total correct answers. Which team got more correct answers altogether?

Find \(p\) and \(q\) such that \(p\begin{pmatrix}3 & 1 \\ 0 & 2\end{pmatrix} + q\begin{pmatrix}1 & 2 \\ -3 & 1\end{pmatrix} = \begin{pmatrix}9 & 8 \\ -9 & 7\end{pmatrix}\).

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