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Exam-Style Questions on MatricesProblems on Matrices adapted from questions set in previous Mathematics exams. |
1. | IB Standard |
A furniture shop sells four items: chairs, beds, stools and tables. Over one week the manager recorded how many of each item were sold on each weekday. The data are arranged in the \(5 \times 4\) matrix \(A\) where rows correspond to Monday, Tuesday, Wednesday, Thursday and Friday in that order, and the columns are chairs, beds, stools, tables in that order.
$$ A \;=\; \begin{bmatrix} 6 & 2 & 5 & 3 \\ 4 & 1 & 3 & 2 \\ 7 & 3 & 4 & 4 \\ 5 & 2 & 2 & 1 \\ 9 & 4 & 6 & 5 \end{bmatrix} $$
The prices of the four items, in British pounds, are stored in the \(4 \times 1\) column matrix \(P\) below, using the same item order as the columns of \(A\).
$$ P \;=\; \begin{bmatrix} 55 \\[2pt] 420 \\[2pt] 30 \\[2pt] 160 \end{bmatrix} \qquad\text{(entries are in pounds)} $$
(a) By performing the matrix multiplication \(T = A P\), find the \(5 \times 1\) matrix \(T\) that gives the total amount taken, in pounds, on each weekday. State clearly which row of \(T\) corresponds to which day.
(b) Use your result for \(T\) to identify the weekday with the greatest takings.
(c) Hence find the total amount taken for the whole week.
2. | IB Standard |
Let \(A\) = \( \begin{pmatrix} 1 & 3 & 2 \\ 3 & 0 & 1 \\ 1 & 1 & 2 \\ \end{pmatrix} \) and \(B\) = \( \begin{pmatrix} 7 \\ 7 \\ 5 \\ \end{pmatrix} \)
(a) Write down \(A^{-1}\)
(b) Find \(X\) if \(AX=B\)
3. | IB Standard |
If the equations below can be represented as the matrix equation \(AX=B\), where \(X=\begin{pmatrix}x\\y\\z\end{pmatrix}\)
$$x-2y-z=2$$ $$x+y=1$$ $$-2x+y-3z=-23$$(a) What is the matrix \(A\) ?
(b) What is the matrix \(B\) ?
(c) Find the matrix \(A^{-1}\).
(d) Use your answers to the previous three parts of this question to find the values of \(x,y\) and \(z\).
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