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: The inverse 'undoes' a matrix transformation. It exists only when \det(A) \neq 0, meaning the transformation does not collapse any dimension.

Common stuck point: Not every matrix has an inverse. If \det(A) = 0, the matrix is singular and A^{-1} does not exist. Always check the determinant first.

Sense of Study hint: First compute the determinant — if it is zero, stop immediately because the inverse does not exist. For a 2 \times 2 matrix, swap the diagonal entries, negate the off-diagonal entries, and divide everything by the determinant. Finally, verify your result by checking that AA^{-1} = I.

Worked Examples

Example 1

medium
Find the inverse of A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}.

Solution

  1. 1
    Step 1: \det(A) = 2(3) - 1(5) = 6 - 5 = 1.
  2. 2
    Step 2: Swap a and d, negate b and c: \frac{1}{1}\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}.
  3. 3
    Step 3: A^{-1} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}.
  4. 4
    Check: AA^{-1} = \begin{bmatrix} 6-5 & -2+2 \\ 15-15 & -5+6 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ✓

Answer

\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}
The 2 \times 2 inverse formula swaps the diagonal entries, negates the off-diagonal entries, and divides by the determinant. The check AA^{-1} = I confirms correctness.

Example 2

easy
Find the inverse of \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}.

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.
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Background Knowledge

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

determinantmatrix multiplication