Practice Inverse Matrix 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 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 $A \cdot A^{-1} = I$ .

Showing a random 20 of 50 problems.

Example 1

easy

Find the inverse of

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

Example 2

medium

Compute the inverse of the rotation matrix

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

Example 3

hard

A

and

B

are invertible

n\times n

matrices, simplify

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

Example 4

medium

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

, find

A

Example 5

hard

Use the inverse matrix to solve

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

Example 6

challenge

2 \times 2

matrix satisfies

A^2 = A

and

A \neq I

. Can

A

be invertible? Explain.

Example 7

easy

Find the inverse of

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

Example 8

easy

Compute the determinant needed to invert

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

Example 9

medium

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

, find

A

Example 10

easy

Does

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

have an inverse?

Example 11

hard

Find

A^{-1}

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

, or show it doesn't exist.

Example 12

easy

What must

AA^{-1}

equal?

Example 13

medium

\det A = 5

, what is

\det(A^{-1})

Example 14

medium

Find the inverse of

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

Example 15

easy

Is the inverse of an invertible matrix unique?

Example 16

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.

Example 17

hard

For an orthogonal matrix

Q

(i.e.,

Q^T Q = I

), what is

Q^{-1}

Example 18

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 19

easy

What is the determinant condition for a matrix to be invertible?

Example 20

medium

Find the inverse of

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

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Related Concepts

Determinant Matrix Multiplication Solving Systems of Equations with Matrices