Vectors and matrices — ACT practice questions

Vectors and matrices on the ACT Mathematics test covers vector length, matrix equality, and using given entries to find a related value. A student has to compute magnitude from components, recognize when two matrices are the same, and combine named entries after a simple operation. The exam tends to ask for the magnitude of a vector, which matrix equals a given expression, or a value such as a plus b. Answer choices are usually numbers or small matrices. Common traps include swapping components, treating matrix multiplication as commutative, and choosing a matrix with the right entries in the wrong order.

Section: Mathematics 45 questions on the paper
45 practice questions in the app

Sample questions

Question 1

What is the magnitude of the vector $\langle -6, 8 \rangle$ ?

A2
B10
C14
D100

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Answer: B — 10

Magnitude is $\sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ .

Question 2

Which of the following matrices is equal to $3\begin{bmatrix}2 & 0 \\ 0 & 5\end{bmatrix}-\begin{bmatrix}3 & 1 \\ 0 & 2\end{bmatrix}$ ?

A $\begin{bmatrix}3 & -1 \\ 0 & 13\end{bmatrix}$
B $\begin{bmatrix}3 & 1 \\ 0 & 13\end{bmatrix}$
C $\begin{bmatrix}3 & -1 \\ 0 & 17\end{bmatrix}$
D $\begin{bmatrix}9 & -1 \\ 0 & 28\end{bmatrix}$

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Answer: A — $\begin{bmatrix}3 & -1 \\ 0 & 13\end{bmatrix}$

Scale first: $3\begin{bmatrix}2 & 0 \\ 0 & 5\end{bmatrix}=\begin{bmatrix}6 & 0 \\ 0 & 15\end{bmatrix}$ . Subtracting entry by entry gives $6-3=3$ , $0-1=-1$ , $0-0=0$ , and $15-2=13$ .

Question 3

If $\begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}\begin{bmatrix}a \\ b\end{bmatrix}=\begin{bmatrix}7 \\ 3\end{bmatrix}$ , what is the value of $a+b$ ?

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Answer: C — 4

The second row gives $b=3$ . Then $a+2b=7$ , so $a=1$ and $a+b=4$ .

Practice 45 Vectors and matrices questions in the app

Vectors and matrices — ACT practice questions

Sample questions

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Mathematics