Matrix multiplication Problem set A

Work out each product, or say why it does not exist.

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

\(A\) has order 3 × 2, \(B\) has order 2 × 4 and \(C\) has order 4 × 3. Give the order of each product, or say that it does not exist.

1. (a) \(AB\)
2. (b) \(BA\)
3. (c) \(BC\)
4. (d) \(CA\)
5. (e) \(AC\)
6. (f) \(CB\)

\(A = \begin{pmatrix}2 & -1 \\ 3 & 0\end{pmatrix}\) and \(B = \begin{pmatrix}1 & 4 \\ -2 & 5\end{pmatrix}\). Find:

1. (a) \(AB\)
2. (b) \(BA\)
3. (c) \(A^2\)
4. (d) \(A^2 - 3A + 2I\)

Find the unknowns.

1. (a) \(\begin{pmatrix}x & 2 \\ 1 & y\end{pmatrix}\begin{pmatrix}3 \\ -1\end{pmatrix} = \begin{pmatrix}10 \\ 5\end{pmatrix}\)
2. (b) \(\begin{pmatrix}2 & p \\ 1 & 3\end{pmatrix}\begin{pmatrix}1 & 4 \\ q & 2\end{pmatrix} = \begin{pmatrix}-4 & 14 \\ -5 & 10\end{pmatrix}\)

A market stall sells apples, pears and plums by the kilogram. The prices in pounds at two stalls, X and Y, are \(P = \begin{pmatrix}2.40 & 2.60 \\ 4.50 & 4.20 \\ 1.80 & 2.00\end{pmatrix}\) (rows: apples, pears, plums; columns: X, Y). (a) Stall X raises all its prices by 4% and stall Y cuts all its prices by 3%. Write this as a matrix product and find the new prices. (b) Instead, apples go up 10%, pears stay the same and plums go down 5%. Write this as a matrix product and find the new prices.

A cinema shows three films. The numbers of adult and child tickets sold are \(\begin{pmatrix}120 & 45 \\ 80 & 110 \\ 65 & 20\end{pmatrix}\). An adult ticket costs £9.50 and a child ticket costs £6. Use a matrix product to find the takings for each film.

In a survey, the proportions of Year 12 and Year 13 students choosing each of three lunch options were \(P = \begin{pmatrix}0.40 & 0.25 \\ 0.35 & 0.45 \\ 0.25 & 0.30\end{pmatrix}\) (columns: Year 12, Year 13). Two schools have \(N = \begin{pmatrix}180 & 200 \\ 160 & 140\end{pmatrix}\) students (rows: Year 12, Year 13; columns: school 1, school 2). Assuming the proportions hold, find the matrix giving the expected number of each lunch in each school, and the total number of the second option needed.

A workshop builds two models of bike, road and trail. This week it builds \(\begin{pmatrix}6 & 4\end{pmatrix}\) (road, trail). The numbers of frames, wheels and gear sets each model needs are \(\begin{pmatrix}1 & 2 & 1 \\ 1 & 2 & 2\end{pmatrix}\) (rows: road, trail). Two shops charge, in euros, \(\begin{pmatrix}120 & 110 \\ 35 & 38 \\ 40 & 36\end{pmatrix}\) (rows: frame, wheel, gear set; columns: shop 1, shop 2). Find the total cost of the parts at each shop, and say which shop is cheaper and by how much.

Find the value of \(k\) for which \(\begin{pmatrix}2 & 1 \\ 0 & 3\end{pmatrix}\) and \(\begin{pmatrix}k & 5 \\ 0 & 4\end{pmatrix}\) commute.

\(A = \begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}\). Find \(A^2\) and \(A^3\), and suggest a formula for \(A^n\).

A student writes \((A + B)^2 = A^2 + 2AB + B^2\) for two square matrices. Explain what is wrong, and write the correct expansion.