Matrices and Systems of Equations: math practice, Grade 12

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Math practice Grade 12 : Matrices and Systems of Equations — Zyro the alien explorer of Planète Maths

20 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

1 Reading a matrix ★★★

Let $M = \begin{bmatrix}4&-2&0\\7&1&9\end{bmatrix}$. (a) Give the size of $M$. (b) Find $m_{12}$ and $m_{23}$.

2 Adding and subtracting ★★★

Let $P = \begin{bmatrix}3&-1\\0&5\end{bmatrix}$ and $Q = \begin{bmatrix}2&4\\-6&1\end{bmatrix}$. Find $P + Q$ and $P - Q$.

3 Scalar multiples ★★★

Compute (a) $-2R$ for $R = \begin{bmatrix}1&-3&2\\0&4&-5\end{bmatrix}$ and (b) $\dfrac{1}{2}S$ for $S = \begin{bmatrix}6&-4\\10&2\end{bmatrix}$.

4 Is the product defined? ★★★

Let $A$ be $2 \times 3$, $B$ be $3 \times 4$, $C$ be $3 \times 1$, and $D$ be $1 \times 3$. For each product, say whether it is defined and give its size: (a) $AB$ (b) $BA$ (c) $CD$ (d) $DC$.

5 Two-by-two determinants ★★★

Compute each determinant: (a) $\begin{bmatrix}5&2\\3&4\end{bmatrix}$ (b) $\begin{bmatrix}-3&6\\1&-2\end{bmatrix}$ (c) $\begin{bmatrix}0&7\\-1&4\end{bmatrix}$. Which matrix has no inverse?

6 Check an inverse ★★★

Show that $\begin{bmatrix}2&-1\\-5&3\end{bmatrix}$ is the inverse of $A = \begin{bmatrix}3&1\\5&2\end{bmatrix}$.

7 Write a system as a matrix equation ★★★

Write $4x + 3y = 11$ and $x - 3y = -1$ in the form $AX = C$, then verify that $(2, 1)$ is a solution.

8 Multiplying in both orders ★★★

Let $A = \begin{bmatrix}1&4\\-2&3\end{bmatrix}$ and $B = \begin{bmatrix}5&0\\2&-1\end{bmatrix}$. Find $AB$ and $BA$. Are they equal?

9 Store revenue ★★★

A school store has two locations. Units sold of pencils, notebooks, and folders: East $30, 12, 20$; West $25, 18, 10$. Prices are $\$0.50$, $\$3.20$, and $\$2.40$. Use a matrix product to find each location’s revenue.

10 A three-by-three determinant ★★★

Find $\det N$ for $N = \begin{bmatrix}1&2&0\\3&-1&4\\2&0&5\end{bmatrix}$.

11 Finding inverses ★★★

Find the inverse of (a) $\begin{bmatrix}4&3\\5&4\end{bmatrix}$ and (b) $\begin{bmatrix}2&6\\1&4\end{bmatrix}$.

12 Solve with an inverse ★★★

Solve $5x + 3y = 11$ and $2x + y = 5$ using the inverse of the coefficient matrix.

13 Gaussian elimination ★★★

Use row operations to solve $x + 2y - z = 3$, $2x + 3y + z = 11$, $3x - y + z = 10$.

14 Cramer’s Rule practice ★★★

Use Cramer’s Rule to solve $2x + 5y = -4$ and $4x - 3y = 18$.

15 Shear then reflection ★★★

Triangle $T$ has vertices $(1, 1)$, $(4, 1)$, $(1, 3)$. It is first sheared by $H = \begin{bmatrix}1&2\\0&1\end{bmatrix}$, then reflected in the x-axis by $F = \begin{bmatrix}1&0\\0&-1\end{bmatrix}$. (a) Find the single matrix for the combined transformation. (b) Find the image of each vertex. (c) Find the area of $T$ and of its image.

16 Rotation and reflection: order matters ★★★

Let $R = \begin{bmatrix}0&-1\\1&0\end{bmatrix}$ (rotation of $90^\circ$ counterclockwise) and $F = \begin{bmatrix}1&0\\0&-1\end{bmatrix}$ (reflection in the x-axis). Find $FR$ and $RF$, and apply each to the point $(2, 5)$. What do you notice?

17 Area with determinants ★★★

Triangle $OPQ$ has $O(0, 0)$, $P(5, 1)$, $Q(2, 4)$. (a) Find its area with a determinant. (b) The matrix $M = \begin{bmatrix}3&0\\1&2\end{bmatrix}$ maps $P$ and $Q$ to $P'$ and $Q'$. Find them and the area of $OP'Q'$. (c) Explain the result using $\det M$.

18 A system with a parameter ★★★

Consider $2x + ky = 5$ and $6x + 9y = 12$. (a) For which value of $k$ does the coefficient determinant equal zero? (b) For that $k$, how many solutions are there? (c) Solve the system for $k = 1$ with Cramer’s Rule.

19 Trail mix prices ★★★

A shop sells almonds, raisins, and cashews by the pound at prices $a$, $r$, and $c$ dollars. An order of $2$ lb almonds, $1$ lb raisins, and $3$ lb cashews costs $\$40$; $1$ lb almonds, $2$ lb raisins, and $1$ lb cashews costs $\$22$; $3$ lb almonds, $1$ lb raisins, and $2$ lb cashews costs $\$38$. Find the price per pound of each item (about $0.45$ kg) by Gaussian elimination.