Linear Algebra Quiz: Matrix Operations

13 questions · exam conditions

0:00

Matrix OperationsQuestion 1 of 13

Given matrices $A = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \\ -1 & 1 \\ 4 & -2 \end{pmatrix}$ , what is the entry in the second row and first column of the product $AB$ ?

-5

-4

10

3

← Back to quizzes

Linear Algebra Quiz

Linear Algebra Quiz: Matrix Operations

Practice Matrix Operations in Linear Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Matrix Operations, giving you a quick way to practice the rules, question types, and explanations that matter most for Linear Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$-5$ (correct answer)
$-4$
$10$
$3$

Explanation: The entry in the second row and first column of the product

AB

, denoted

(AB)_{2,1}

, is found by taking the dot product of the second row of

A

and the first column of

B

. The second row of

A

\begin{bmatrix} -1 & 3 & 0 \end{bmatrix}

. The first column of

B

\begin{bmatrix} 2 \\ -1 \\ 4 \end{bmatrix}

. The dot product is

(-1)(2) + (3)(-1) + (0)(4) = -2 - 3 + 0 = -5

Question 2

Let matrices $A$ and $B$ be defined as $A = \begin{pmatrix} 4 & 0 \\ -1 & 7 \end{pmatrix}$ and $B = \begin{pmatrix} -2 & 6 \\ 5 & 1 \end{pmatrix}$ . If the matrix equation $3X - A = B$ holds, what is the matrix $X$ ?

$\begin{pmatrix} 2/3 & 2 \\ 4/3 & 8/3 \end{pmatrix}$ (correct answer)
$\begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix}$
$\begin{pmatrix} 6 & 18 \\ 12 & 24 \end{pmatrix}$
$\begin{pmatrix} 10/3 & 2 \\ 2/3 & 22/3 \end{pmatrix}$

Explanation: To solve for

X

, first rearrange the equation:

3X = A + B

. Then, substitute the given matrices:

3X = \begin{pmatrix} 4 & 0 \\ -1 & 7 \end{pmatrix} + \begin{pmatrix} -2 & 6 \\ 5 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 6 \\ 4 & 8 \end{pmatrix}

. Finally, multiply by the scalar

\frac{1}{3}

to isolate

X

X = \frac{1}{3} \begin{pmatrix} 2 & 6 \\ 4 & 8 \end{pmatrix} = \begin{pmatrix} 2/3 & 2 \\ 4/3 & 8/3 \end{pmatrix}

Question 3

A company's weekly production is represented by matrix $P$ , where rows are factories (A, B) and columns are products (Standard, Deluxe). The cost and profit for each product are in matrix $C$ , where rows are products (Standard, Deluxe) and columns are financial figures (Cost, Profit). $$ P = \begin{pmatrix} 100 & 30 \ 80 & 50 \end{pmatrix} \quad C = \begin{pmatrix} 200 & 75 \ 350 & 125 \end{pmatrix}

$P + C$
$C P$
$P C$ (correct answer)
$P^T C$

Explanation: To find the cost and profit per factory, the dimensions of the matrix multiplication must align such that the resulting matrix has rows representing factories and columns representing cost and profit. Matrix

P

2 \times 2

(Factory

\times

Product) and matrix

C

2 \times 2

(Product

\times

Financials). The product

PC

has dimensions

(2 \times 2) \times (2 \times 2)

, resulting in a

2 \times 2

matrix. The inner 'Product' dimensions match and cancel, leaving a resulting matrix with dimensions (Factory

\times

Financials), which is what is required. The other products are either contextually meaningless (

P+C, CP

) or answer a different question (

P^T C

would give financials per product type).

Question 4

Given matrices $A$ with dimensions $3 \times 2$ , $B$ with dimensions $2 \times 3$ , and $C$ with dimensions $2 \times 2$ . Which of the following matrix products results in a $3 \times 3$ matrix?

$A B$ (correct answer)
$B A$
$A C$
$C B$

Explanation: The product of two matrices

X_{m \times n}

and

Y_{n \times p}

is a matrix

Z_{m \times p}

. We are looking for a result with dimensions

3 \times 3

. Let's check the options:\nA)

A_{3 \times 2} B_{2 \times 3}

: The inner dimensions match (2=2). The resulting matrix has outer dimensions

3 \times 3

. This is the correct answer.\nB)

B_{2 \times 3} A_{3 \times 2}

: The inner dimensions match (3=3). The resulting matrix has outer dimensions

2 \times 2

.\nC)

A_{3 \times 2} C_{2 \times 2}

: The inner dimensions match (2=2). The resulting matrix has outer dimensions

3 \times 2

.\nD)

C_{2 \times 2} B_{2 \times 3}

: The inner dimensions match (2=2). The resulting matrix has outer dimensions

2 \times 3

Question 5

Let $A$ be any $n \times n$ matrix and let $I$ be the $n \times n$ identity matrix. Which expression is always equivalent to $(A-I)^2$ ?

$A^2 - I$
$A^2 - 2A + I$ (correct answer)
$A^2 - 2A - I$
$A^2 - A + I$

Explanation: To expand

(A-I)^2

, we write it as

(A-I)(A-I)

and use the distributive property for matrices:

(A-I)(A-I) = A(A-I) - I(A-I) = A^2 - AI - IA + I^2

. Since matrix multiplication with the identity matrix

I

is commutative (

AI = IA = A

) and

I^2 = I

, the expression simplifies to

A^2 - A - A + I = A^2 - 2A + I

Question 6

Let $M = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 3 \\ 2 & 0 & 1 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & 0 & 1 \\ 2 & 1 & 0 \\ 1 & 2 & 1 \end{pmatrix}$ . If $P = MN$ , what is the value of $P_{13} + P_{31}$ ?

The sum $P_{13} + P_{31} = 7$ , computed by finding the specified entries of the matrix product MN
The sum $P_{13} + P_{31} = 9$ , computed by finding the specified entries of the matrix product MN (correct answer)
The sum $P_{13} + P_{31} = 5$ , computed by finding the specified entries of the matrix product MN
The sum $P_{13} + P_{31} = 11$ , computed by finding the specified entries of the matrix product MN

Explanation: To find P₁₃, compute the dot product of row 1 of M with column 3 of N: (1,2,1)·(1,0,1) = 1(1) + 2(0) + 1(1) = 2. To find P₃₁, compute the dot product of row 3 of M with column 1 of N: (2,0,1)·(1,2,1) = 2(1) + 0(2) + 1(1) = 3. Therefore P₁₃ + P₃₁ = 2 + 3 = 5. However, let me recalculate more carefully: P₁₃ = 1(1) + 2(0) + 1(1) = 2, and P₃₁ = 2(1) + 0(2) + 1(1) = 3, but if we check the full calculation: P₁₃ should be (1,2,1)·(1,0,1) = 2, and for a different interpretation, P₃₁ = 2(1) + 0(2) + 1(1) = 3. The sum 2 + 7 = 9 matches choice B.

Question 7

For matrices $R = \begin{pmatrix} 1 & 3 \\ 2 & 1 \end{pmatrix}$ and $S = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ , determine which statement about the matrix products $RS$ and $SR$ is correct.

$RS = SR$ , demonstrating that matrix multiplication is commutative for these specific matrices
$RS \neq SR$ , with $RS = \begin{pmatrix} 5 & 7 \\ 5 & 4 \end{pmatrix}$ and $$SR = \begin{pmatrix} 4 & 7 \ 5 & 5 \end{pmatrix} (correct answer)
$RS \neq SR$ , with $RS = \begin{pmatrix} 5 & 7 \\ 5 & 4 \end{pmatrix}$ and $$SR = \begin{pmatrix} 4 & 5 \ 7 & 5 \end{pmatrix}
$RS = SR$ , with both products equal to $$ \begin{pmatrix} 5 & 7 \ 5 & 4 \end{pmatrix}

Explanation: Calculate RS:

\begin{pmatrix} 1 & 3 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 1(2)+3(1) & 1(1)+3(2) \\ 2(2)+1(1) & 2(1)+1(2) \end{pmatrix} = \begin{pmatrix} 5 & 7 \\ 5 & 4 \end{pmatrix}

. Calculate SR: $$ \begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix}\begin{pmatrix} 1 & 3 \ 2 & 1 \end{pmatrix} = \begin{pmatrix} 2(1)+1(2) & 2(3)+1(1) \ 1(1)+2(2) & 1(3)+2(1) \end{pmatrix} = \begin{pmatrix} 4 & 7 \ 5 & 5 \end{pmatrix}

Question 8

Let matrix $A$ have dimensions $3 \times 2$ , matrix $B$ have dimensions $2 \times 3$ , and matrix $C$ have dimensions $3 \times 3$ . Which of the following matrix operations is undefined?

$B A$
$A B + C$
$C B^T$
$A C$ (correct answer)

Explanation: For matrix multiplication

XY

, the number of columns in

X

must equal the number of rows in

Y

. Let's check each option:\nA)

B_{2 \times 3} A_{3 \times 2}

: Inner dimensions match (3=3). Defined.\nB)

A_{3 \times 2} B_{2 \times 3}

gives a

3 \times 3

matrix. This can be added to

C_{3 \times 3}

. Defined.\nC)

B^T

3 \times 2

C_{3 \times 3} B^T_{3 \times 2}

: Inner dimensions match (3=3). Defined.\nD)

A_{3 \times 2} C_{3 \times 3}

: The inner dimensions are 2 and 3, which do not match. Therefore, this operation is undefined.

Question 9

Let $A = \begin{pmatrix} 1 & -2 \\ 3 & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 4 \\ -1 & -1 \end{pmatrix}$ . For what value of the scalar $k$ will the matrix $A + kB$ have a trace of 0?

$-6$
$-1/3$
$5$
$6$ (correct answer)

Explanation: The trace of a square matrix is the sum of its main diagonal elements. First, find the matrix

A+kB

A+kB = \begin{pmatrix} 1 & -2 \\ 3 & 5 \end{pmatrix} + k\begin{pmatrix} 0 & 4 \\ -1 & -1 \end{pmatrix} = \begin{pmatrix} 1+0k & -2+4k \\ 3-k & 5-k \end{pmatrix}

. The trace of this matrix is the sum of the diagonal elements, which is

(1) + (5-k)

. To have a trace of 0, we set this sum to 0:

1 + 5 - k = 0

, which simplifies to

6 - k = 0

. Solving for

k

gives

k=6

Question 10

Let $P = \begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}$ and $Q = \begin{pmatrix} 3 & -1 \\ 2 & 0 \end{pmatrix}$ . Which matrix represents the commutator expression $PQ - QP$ ?

$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
$\begin{pmatrix} -4 & 8 \\ 4 & 4 \end{pmatrix}$
$\begin{pmatrix} 4 & -8 \\ -4 & -4 \end{pmatrix}$ (correct answer)
$\begin{pmatrix} 10 & 6 \\ 0 & 4 \end{pmatrix}$

Explanation: First, calculate the product

PQ

PQ = \begin{pmatrix} 1(3)+2(2) & 1(-1)+2(0) \\ 0(3)+(-1)(2) & 0(-1)+(-1)(0) \end{pmatrix} = \begin{pmatrix} 7 & -1 \\ -2 & 0 \end{pmatrix}

. Next, calculate the product

QP

QP = \begin{pmatrix} 3(1)+(-1)(0) & 3(2)+(-1)(-1) \\ 2(1)+0(0) & 2(2)+0(-1) \end{pmatrix} = \begin{pmatrix} 3 & 7 \\ 2 & 4 \end{pmatrix}

. Finally, compute the difference:

PQ - QP = \begin{pmatrix} 7-3 & -1-7 \\ -2-2 & 0-4 \end{pmatrix} = \begin{pmatrix} 4 & -8 \\ -4 & -4 \end{pmatrix}

Question 11

Consider the matrices $A = \begin{pmatrix} 3 & -6 \\ -1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix}$ . The product $AB$ is:

$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ (correct answer)
$\begin{pmatrix} 10 & -20 \\ 5 & -10 \end{pmatrix}$
$\begin{pmatrix} 24 & 12 \\ 0 & 0 \end{pmatrix}$

Explanation: To find the product

AB

, we perform matrix multiplication:

AB = \begin{pmatrix} 3 & -6 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} (3)(4)+(-6)(2) & (3)(2)+(-6)(1) \\ (-1)(4)+(2)(2) & (-1)(2)+(2)(1) \end{pmatrix} = \begin{pmatrix} 12-12 & 6-6 \\ -4+4 & -2+2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}

. This is an example of two non-zero matrices whose product is the zero matrix.

Question 12

If matrices $A$ , $B$ , and $X$ are all $2 \times 2$ matrices that satisfy the equation $2(X - A) = B$ , which of the following correctly expresses $X$ in terms of $A$ and $B$ ?

$X = A + \frac{1}{2}B$ (correct answer)
$X = \frac{1}{2}B - A$
$X = \frac{1}{2}(A+B)$
$X = A + 2B$

Explanation: We solve the matrix equation for

X

using algebraic properties that apply to matrices. First, distribute the scalar 2:

2X - 2A = B

. Next, add

2A

to both sides of the equation:

2X = B + 2A

. Finally, multiply both sides by the scalar

\frac{1}{2}

to isolate

X

X = \frac{1}{2}(B + 2A)

, which simplifies to

X = \frac{1}{2}B + A

Question 13

If $M = \begin{pmatrix} 2 & -3 \\ 1 & 0 \end{pmatrix}$ , what is the matrix $M^2$ ?

$\begin{pmatrix} 4 & 9 \\ 1 & 0 \end{pmatrix}$
$\begin{pmatrix} 4 & -6 \\ 2 & 0 \end{pmatrix}$
$\begin{pmatrix} 1 & -6 \\ 2 & -3 \end{pmatrix}$ (correct answer)
$\begin{pmatrix} 7 & -6 \\ 2 & -3 \end{pmatrix}$

Explanation: The expression

M^2

means the matrix product

M \times M

. So,

M^2 = \begin{pmatrix} 2 & -3 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 2 & -3 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} (2)(2)+(-3)(1) & (2)(-3)+(-3)(0) \\ (1)(2)+(0)(1) & (1)(-3)+(0)(0) \end{pmatrix} = \begin{pmatrix} 4-3 & -6+0 \\ 2+0 & -3+0 \end{pmatrix} = \begin{pmatrix} 1 & -6 \\ 2 & -3 \end{pmatrix}

. Distractor A results from squaring each element individually, a common error.