Matrix multiplication Problem set B

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

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

\(A\) has order 4 × 3, \(B\) has order 3 × 2 and \(C\) has order 2 × 4. 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) \(CB\)

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

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

Find the unknowns.

1. (a) \(\begin{pmatrix}x & 3 \\ 2 & y\end{pmatrix}\begin{pmatrix}2 \\ -1\end{pmatrix} = \begin{pmatrix}7 \\ 5\end{pmatrix}\)
2. (b) \(\begin{pmatrix}3 & p \\ 2 & 1\end{pmatrix}\begin{pmatrix}2 & 1 \\ q & 4\end{pmatrix} = \begin{pmatrix}0 & 11 \\ 1 & 6\end{pmatrix}\)

A garden centre sells tulip, rose and lily bulbs by the dozen. The prices in euros at its two branches, N and S, are \(P = \begin{pmatrix}3.2 & 3.5 \\ 5 & 4.8 \\ 2.4 & 2.6\end{pmatrix}\) (rows: tulips, roses, lilies; columns: N, S). (a) Branch N raises all its prices by 5% and branch S cuts all its prices by 2%. Write this as a matrix product and find the new prices. (b) Instead, tulips go up 8%, roses stay the same and lilies go down 10%. Write this as a matrix product and find the new prices.

A museum runs three exhibitions. The numbers of adult and child tickets sold are \(\begin{pmatrix}110 & 35 \\ 90 & 60 \\ 75 & 40\end{pmatrix}\). An adult ticket costs €11.50 and a child ticket costs €7. Use a matrix product to find the takings for each exhibition.

In a survey, the proportions of Grade 11 and Grade 12 students choosing each of three after-school clubs were \(P = \begin{pmatrix}0.3 & 0.2 \\ 0.45 & 0.5 \\ 0.25 & 0.3\end{pmatrix}\) (columns: Grade 11, Grade 12). Two campuses have \(N = \begin{pmatrix}160 & 240 \\ 200 & 120\end{pmatrix}\) students (rows: Grade 11, Grade 12; columns: campus 1, campus 2). Assuming the proportions hold, find the matrix giving the expected number of students in each club on each campus, and the total number of places needed in the second club.

A bakery makes two cakes, lemon and chocolate. In one day it makes \(\begin{pmatrix}30 & 20\end{pmatrix}\) (lemon, chocolate). The packs of flour, sugar and butter each cake needs are \(\begin{pmatrix}3 & 2 & 1 \\ 2 & 2 & 3\end{pmatrix}\) (rows: lemon, chocolate). Two wholesalers charge, in euros per pack, \(\begin{pmatrix}0.9 & 0.85 \\ 1.2 & 1.25 \\ 1.5 & 1.4\end{pmatrix}\) (rows: flour, sugar, butter; columns: wholesaler 1, wholesaler 2). Find the cost of the day's packs from each wholesaler, and say which is cheaper and by how much.

Find the value of \(k\) for which \(\begin{pmatrix}2 & 1 \\ 0 & 4\end{pmatrix}\) and \(\begin{pmatrix}k & 6 \\ 0 & 3\end{pmatrix}\) commute.

\(A = \begin{pmatrix}1 & 3 \\ 0 & 1\end{pmatrix}\). 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.