Practice Determinant in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The determinant is a scalar value computed from a square matrix that encodes important geometric and algebraic information. For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the determinant is $ad - bc$ . A nonzero determinant means the matrix is invertible.

The determinant measures how a matrix scales area (in 2D) or volume (in 3D). If $\det(A) = 3$ , the transformation described by $A$ triples all areas. If $\det(A) = 0$ , the transformation collapses space into a lower dimension (like squishing a plane into a line), which is why the matrix has no inverse.

Showing a random 20 of 50 problems.

Example 1

medium

Find the determinant of

\begin{pmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}

Example 2

challenge

2 \times 2

matrix

A

has

\det(A) = 7

. What is

\det(A^{-1})

and why?

Example 3

medium

Find

k

so that

\begin{bmatrix} 3 & k \\ 6 & 8 \end{bmatrix}

is singular.

Example 4

medium

A

2 \times 2

with

\det(A) = -5

, compute

\det(A^3)

Example 5

easy

Find the determinant of

\begin{pmatrix} 3 & 6 \\ 1 & 2 \end{pmatrix}

Example 6

medium

2\times 2

matrix with two identical rows has determinant ____.

Example 7

medium

If a

2 \times 2

matrix has

\det = 0

, what can you say about its rows?

Example 8

challenge

For

2 \times 2

matrices with

\det(A) = 3

and

\det(B) = 4

, find

\det(AB)

Example 9

easy

Find

\det\begin{bmatrix} 4 & 7 \\ 1 & 2 \end{bmatrix}

Example 10

medium

Compute

\det\begin{bmatrix} 1 & 2 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 5 \end{bmatrix}

Example 11

easy

The determinant of a diagonal matrix

\begin{bmatrix} 7 & 0 \\ 0 & 3 \end{bmatrix}

equals ____.

Example 12

hard

Use cofactor expansion along column 2 to compute

\det\begin{bmatrix} 3 & 0 & 2 \\ 1 & 5 & 4 \\ 2 & 0 & 1 \end{bmatrix}

Example 13

easy

Find the determinant of the identity

\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Example 14

medium

Swapping the two rows of a

2 \times 2

matrix changes the determinant in what way?

Example 15

easy

Find

\det\begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix}

Example 16

easy

What is

\det\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Example 17

medium

Find the determinant of

\begin{pmatrix} 6 & 4 \\ 3 & 5 \end{pmatrix}

Example 18

easy

Compute

\det\begin{bmatrix} 9 & 4 \\ 6 & 3 \end{bmatrix}

and decide whether the matrix is invertible.

Example 19

medium

Compute the

3 \times 3

determinant of

\begin{pmatrix} 2 & 0 & 1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}

by cofactor expansion along row 1.

Example 20

hard

The triangle with vertices

(0,0)

(4,1)

, and

(2,5)

has what area?

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Related Concepts

Matrix Definition Matrix Multiplication Inverse Matrix Solving Systems of Equations with Matrices Cross Product