Determinant Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Determinant.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The determinant is a single number measuring whether a square matrix is invertible and how it scales area or volume.

Common stuck point: The procedure for determinant is the easy part; the trap is computing $ad+bc$ instead of $ad-bc$ . Asking "Is the matrix square, and am I asking whether it is invertible or how it scales area?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the matrix square, and am I asking whether it is invertible or how it scales area?

Worked Examples

Example 1

easy

Find

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

Answer

10

First step

Step 1: Apply formula:

\det = ad - bc

where

a=3, b=1, c=2, d=4

Full solution

2
Step 2: $\det = 3(4) - 1(2) = 12 - 2 = 10$ .
3
Check: Since $\det \neq 0$ , the matrix is invertible ✓

The

2 \times 2

determinant is computed as

ad - bc

(product of main diagonal minus product of anti-diagonal). A nonzero determinant means the matrix is invertible.

Example 2

hard

Evaluate

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

by expanding along the first row.

Example 3

easy

Compute

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

and decide whether the matrix is invertible.

Example 4

medium

Compute

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

Example 5

hard

Use cofactor expansion along column 2 to compute

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find

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

Example 2

medium

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

invertible?

Example 3

easy

Find the determinant of

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

Example 4

easy

Find the determinant of

\begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}

Example 5

easy

Find the determinant of the identity

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

Example 6

easy

Find the determinant of

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

Example 7

easy

Find the determinant of

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

Example 8

easy

Find the determinant of

\begin{pmatrix} 5 & 2 \\ 1 & 3 \end{pmatrix}

Example 9

easy

Is the matrix

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

invertible?

Example 10

easy

Find the determinant of

\begin{pmatrix} 7 & 0 \\ 0 & 0 \end{pmatrix}

Example 11

medium

Find the determinant of

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

Example 12

medium

For what value of

k

\begin{pmatrix} k & 2 \\ 8 & k \end{pmatrix}

singular?

Example 13

medium

Find the determinant of

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

Example 14

medium

Find the determinant of

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

Example 15

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 16

medium

\det(A) = 5

for a

2 \times 2

matrix, what is

\det(2A)

Example 17

medium

Find the determinant of

\begin{pmatrix} a & b \\ a & b \end{pmatrix}

Example 18

medium

Find the determinant of

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

Example 19

medium

If a

2 \times 2

matrix has

\det = 0

, what can you say about its rows?

Example 20

challenge

Find all

k

so that

\begin{pmatrix} 1 & k \\ k & 4 \end{pmatrix}

has determinant equal to 3.

Example 21

challenge

2 \times 2

matrix

A

has

\det(A) = 7

. What is

\det(A^{-1})

and why?

Example 22

challenge

For

2 \times 2

matrices with

\det(A) = 3

and

\det(B) = 4

, find

\det(AB)

Example 23

easy

Find

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

Example 24

easy

Find

\det\begin{bmatrix} 6 & 5 \\ 2 & 3 \end{bmatrix}

Example 25

easy

Compute

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

Example 26

easy

\begin{bmatrix} 1 & 5 \\ 0 & 0 \end{bmatrix}

invertible?

Example 27

medium

Find all

x

so that

\det\begin{bmatrix} x & 2 \\ 3 & x \end{bmatrix} = 10

Example 28

medium

Find

k

so that

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

is singular.

Example 29

medium

\det(A) = -3

for a

3 \times 3

matrix, find

\det(2A)

Example 30

medium

Compute

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

by cofactor expansion along the first row.

Example 31

medium

\det(A) = 4

, what is

\det(A^T)

Example 32

medium

A

2 \times 2

with

\det(A) = -5

, compute

\det(A^3)

Example 33

medium

Swapping the two rows of a

2 \times 2

matrix changes the determinant in what way?

Example 34

medium

The vectors

(2,3)

and

(5,1)

form a parallelogram. Find its area.

Example 35

hard

Compute

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

Example 36

hard

A

and

B

are

3\times 3

with

\det(A) = 2

and

\det(B) = -4

, find

\det(A^2 B^{-1})

Example 37

hard

For which values of

\lambda

\begin{bmatrix} 4-\lambda & 1 \\ 2 & 3-\lambda \end{bmatrix}

singular?

Example 38

hard

The triangle with vertices

(0,0)

(4,1)

, and

(2,5)

has what area?

Example 39

hard

If the rows of a

3\times 3

matrix

A

are linearly dependent, what is

\det(A)

Example 40

hard

Use Cramer's rule to solve

\begin{cases} 2x + y = 5 \\ x - 3y = -8 \end{cases}

Example 41

challenge

Show that for any invertible

n\times n

matrix

A

\det(A)\det(A^{-1}) = 1

Example 42

challenge

Find all

k

so that the system

\begin{cases} kx + y = 0 \\ x + ky = 0 \end{cases}

has a nontrivial solution.

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Related Concepts

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

Background Knowledge

These ideas may be useful before you work through the harder examples.

matrix definitionmatrix multiplication