Inverse Matrix Examples in Math

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

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 inverse of a square matrix $A$ , written $A^{-1}$ , is the unique matrix such that $AA^{-1} = A^{-1}A = I$ (the identity matrix). A matrix has an inverse if and only if its determinant is nonzero.

If matrix $A$ represents a transformation (like rotating 30 degrees), then $A^{-1}$ undoes that transformation (rotating $-30$ degrees). Multiplying by the inverse is the matrix equivalent of dividing. Just as $5 \times \frac{1}{5} = 1$ , we have $A \cdot A^{-1} = I$ .

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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: $A^{-1}$ is the unique matrix with $AA^{-1}=I$ , and it exists only when $\det A\neq0$ .

Common stuck point: The procedure for inverse matrix is the easy part; the trap is forgetting the $\frac{1}{ad-bc}$ factor. Asking "Is the matrix square with nonzero determinant, so an undo-matrix $A^{-1}$ exists?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the matrix square with nonzero determinant, so an undo-matrix $A^{-1}$ exists?

Worked Examples

Example 1

medium

Find the inverse of

A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}

Answer

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

First step

Step 1:

\det(A) = 2(3) - 1(5) = 6 - 5 = 1

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Example 2

easy

Find the inverse of

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

Example 3

medium

Find the inverse of

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

Example 4

medium

Compute the inverse of the rotation matrix

R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Example 5

hard

Find the inverse of

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

using row reduction.

Example 6

hard

Use the inverse matrix to solve

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

Example 7

challenge

Show that if

A

is a

2\times 2

matrix satisfying

A^2 - 5A + 6I = 0

, then

A

is invertible and find

A^{-1}

in terms of

A

and

I

Practice Problems

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

Example 1

easy

Find the inverse of

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

Example 2

hard

Does

\begin{bmatrix} 6 & 3 \\ 4 & 2 \end{bmatrix}

have an inverse? Explain.

Example 3

easy

What is the inverse of the identity matrix

I

Example 4

easy

Does

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

have an inverse?

Example 5

easy

Compute the determinant needed to invert

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

Example 6

easy

In the

2 \times 2

inverse formula, what happens to entries

b

and

c

Example 7

easy

Find the inverse of

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

Example 8

easy

What must

AA^{-1}

equal?

Example 9

easy

Why is

A^{-1} = \frac{1}{A}

wrong?

Example 10

easy

Is the inverse of an invertible matrix unique?

Example 11

medium

Find the inverse of

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

Example 12

medium

Find the inverse of

\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}

Example 13

medium

Verify that

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

and

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

are inverses by computing their product.

Example 14

medium

Find the inverse of

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

Example 15

medium

Solve

Ax = b

where

A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}

and

b = \begin{pmatrix} 3 \\ 8 \end{pmatrix}

using

A^{-1}

Example 16

medium

A^{-1} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}

, find

A

Example 17

medium

For what value of

k

does

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

fail to have an inverse?

Example 18

medium

Find the inverse of

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

Example 19

medium

Find the inverse of

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

Example 20

challenge

Find the inverse of

\begin{pmatrix} 1 & 2 \\ 3 & 7 \end{pmatrix}

and use it to solve

\begin{cases} x + 2y = 5 \\ 3x + 7y = 18 \end{cases}

Example 21

challenge

Show that

(AB)^{-1} = B^{-1}A^{-1}

, not

A^{-1}B^{-1}

, by reasoning about order.

Example 22

challenge

2 \times 2

matrix satisfies

A^2 = A

and

A \neq I

. Can

A

be invertible? Explain.

Example 23

easy

Find the inverse of

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

Example 24

easy

Compute the inverse of

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

Example 25

easy

Find the inverse of

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

Example 26

easy

What does

A \cdot A^{-1}

equal for an invertible matrix

A

Example 27

medium

For what value of

k

does

\begin{pmatrix} k & 4 \\ 2 & 6 \end{pmatrix}

fail to be invertible?

Example 28

medium

Solve

Ax = b

using

A^{-1}

, where

A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}

and

b = \begin{pmatrix} 5 \\ 3 \end{pmatrix}

Example 29

medium

Find the inverse of

\begin{pmatrix} 5 & 2 \\ 8 & 3 \end{pmatrix}

Example 30

medium

Find the inverse of the diagonal matrix

\begin{pmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix}

Example 31

medium

Verify that

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

and

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

are inverses by computing their product.

Example 32

medium

Find

A^{-1}

for

A = \begin{pmatrix} 1 & 3 \\ 2 & 7 \end{pmatrix}

Example 33

medium

A^{-1} = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}

, find

A

Example 34

medium

Does

\begin{pmatrix} 2 & 4 \\ 3 & 6 \end{pmatrix}

have an inverse? Why?

Example 35

hard

A

and

B

are invertible

n\times n

matrices, simplify

(B A^{-1})^{-1}

Example 36

hard

Find

A^{-1}

A = \begin{pmatrix} 4 & 6 \\ 2 & 3 \end{pmatrix}

, or show it doesn't exist.

Example 37

hard

For invertible matrix

A

, prove that

(A^T)^{-1} = (A^{-1})^T

Example 38

hard

A

is a

2\times 2

matrix with

A^2 = I

and

A \ne \pm I

, is

A

invertible?

Example 39

hard

For an orthogonal matrix

Q

(i.e.,

Q^T Q = I

), what is

Q^{-1}

Example 40

challenge

A

is invertible and

A + A^{-1} = 3I

for a

2\times 2

matrix

A

, find

\det A

given that the eigenvalues of

A

are real.

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Related Concepts

Determinant Matrix Multiplication Solving Systems of Equations with Matrices

Background Knowledge

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

determinantmatrix multiplication