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 AA, written Aโˆ’1A^{-1}, is the unique matrix such that AAโˆ’1=Aโˆ’1A=IAA^{-1} = A^{-1}A = I (the identity matrix). A matrix has an inverse if and only if its determinant is nonzero.

If matrix AA represents a transformation (like rotating 30 degrees), then Aโˆ’1A^{-1} undoes that transformation (rotating โˆ’30-30 degrees). Multiplying by the inverse is the matrix equivalent of dividing. Just as 5ร—15=15 \times \frac{1}{5} = 1, we have Aโ‹…Aโˆ’1=IA \cdot A^{-1} = I.

Showing a random 20 of 50 problems.

Example 1

easy
Find the inverse of [4131]\begin{bmatrix} 4 & 1 \\ 3 & 1 \end{bmatrix}.

Example 2

medium
Compute the inverse of the rotation matrix R(ฮธ)=(cosโกฮธโˆ’sinโกฮธsinโกฮธcosโกฮธ)R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}.

Example 3

hard
If AA and BB are invertible nร—nn\times n matrices, simplify (BAโˆ’1)โˆ’1(B A^{-1})^{-1}.

Example 4

medium
If Aโˆ’1=(2111)A^{-1} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}, find AA.

Example 5

hard
Use the inverse matrix to solve {2x+3y=8x+2y=5\begin{cases} 2x + 3y = 8 \\ x + 2y = 5 \end{cases}.

Example 6

challenge
A 2ร—22 \times 2 matrix satisfies A2=AA^2 = A and Aโ‰ IA \neq I. Can AA be invertible? Explain.

Example 7

easy
Find the inverse of (2004)\begin{pmatrix} 2 & 0 \\ 0 & 4 \end{pmatrix}.

Example 8

easy
Compute the determinant needed to invert (4131)\begin{pmatrix} 4 & 1 \\ 3 & 1 \end{pmatrix}.

Example 9

medium
If Aโˆ’1=(2312)A^{-1} = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}, find AA.

Example 10

easy
Does (1224)\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} have an inverse?

Example 11

hard
Find Aโˆ’1A^{-1} if A=(4623)A = \begin{pmatrix} 4 & 6 \\ 2 & 3 \end{pmatrix}, or show it doesn't exist.

Example 12

easy
What must AAโˆ’1AA^{-1} equal?

Example 13

medium
If detโกA=5\det A = 5, what is detโก(Aโˆ’1)\det(A^{-1})?

Example 14

medium
Find the inverse of (3512)\begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}.

Example 15

easy
Is the inverse of an invertible matrix unique?

Example 16

challenge
If AA is invertible and A+Aโˆ’1=3IA + A^{-1} = 3I for a 2ร—22\times 2 matrix AA, find detโกA\det A given that the eigenvalues of AA are real.

Example 17

hard
For an orthogonal matrix QQ (i.e., QTQ=IQ^T Q = I), what is Qโˆ’1Q^{-1}?

Example 18

medium
Verify that (2111)\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} and (1โˆ’1โˆ’12)\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 (2513)\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}.
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