How to Find the Inverse of a Matrix (2×2 and 3×3)

The inverse of a square matrix $A$ is the matrix $A^{-1}$ with $AA^{-1} = I$, the identity. It exists only when the determinant is not zero. For a 2×2 matrix there is a quick formula; for larger matrices, row-reduce $[A \mid I]$ until the left side becomes $I$.

The 2×2 formula

Swap the diagonal, negate the other two entries, and divide by the determinant.

Worked example 1

$A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$. The determinant is $2 \times 3 - 1 \times 5 = 1$, so

Check: $\begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ ✓

Larger matrices: Gauss–Jordan elimination

1. Write $A$ next to the identity: $[A \mid I]$.
2. Use row operations (swap rows, scale a row, add a multiple of one row to another) to turn the left side into $I$.
3. The right side is now $A^{-1}$.

Worked example 2: a 3×3 matrix

Its determinant is 1, which is why the inverse has whole-number entries.

What inverses are used for

Solving $A\mathbf{x} = \mathbf{b}$: multiply both sides by $A^{-1}$ to get $\mathbf{x} = A^{-1}\mathbf{b}$. This is the matrix version of dividing.

Common mistakes

• Negating the diagonal instead of the off-diagonal in the 2×2 formula.
• Forgetting to divide by the determinant.
• Assuming $AB = BA$: matrix multiplication order matters.

Practice questions

1. Inverse of $\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$?
2. Does $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$ have an inverse?
3. What is $I^{-1}$?

Answers: 1) $\frac{1}{10}\begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix}$ 2) No, the determinant is 0 3) $I$

The matrix calculator shows every row operation with exact fractions and checks $AA^{-1} = I$. Find determinants with the determinant calculator, explained in how to find the determinant.

Frequently asked questions

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