Inverse of a Matrix — Definition, Formula & Examples

Q: What is the difference between the inverse and the transpose of a matrix?

The inverse $A^{-1}$ undoes multiplication by $A$, meaning $AA^{-1} = I$. The transpose $A^T$ simply flips the matrix over its main diagonal, swapping rows and columns. Every matrix has a transpose, but only square matrices with nonzero determinants have inverses. These are completely different operations.

Q: Why can't non-square matrices have inverses?

A true (two-sided) inverse requires $AA^{-1} = A^{-1}A = I$, which demands that both products be defined and result in identity matrices. This is only possible when $A$ is square. Non-square matrices can have one-sided inverses (left or right), but not a full inverse.

Inverse of a Matrix
Matrix Inverse
Multiplicative Inverse of a Matrix

For a square matrix A, the inverse is written A^-1. When A is multiplied by A^-1 the result is the identity matrix I. Non-square matrices do not have inverses.

Note: Not all square matrices have inverses. A square matrix which has an inverse is called invertible or nonsingular, and a square matrix without an inverse is called noninvertible or singular.

AA^-1 = A^-1A = I

Two panels: general 2×2 formula A⁻¹ = (1/(ad−bc)) · [[d,−b],[−c,a]], and a worked example with A = [[4,3],[3,2]], det A = −1, A⁻¹ = [[−2,3],[3,−4]], verified A·A⁻¹ = I.

Here are three ways to find the inverse of a matrix:

1. Shortcut for 2x2 matrices

For , the inverse can be found using this formula:

Example:

2.Augmented matrix method

Use Gauss-Jordan elimination to transform [ A | I ] into [ I | A^-1 ].

Example: The following steps result in .

so we see that .

3. Adjoint method

A^-1 = (adjoint of A) or A^-1 = (cofactor matrix of A)^T

Example: The following steps result in A^-1 for .

The cofactor matrix for A is , so the adjoint is . Since det A = 22, we get

See also

Determinant of a matrix, cofactor

Key Formula

A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \quad \text{for } A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Where:

$A$ = A 2×2 square matrix
$a, b, c, d$ = The entries of the 2×2 matrix
$ad - bc$ = The determinant of A (must be nonzero for the inverse to exist)
$A^{-1}$ = The inverse matrix such that AA^{-1} = A^{-1}A = I

Worked Example

Problem: Find the inverse of the 2×2 matrix A = [[4, 7], [2, 6]].

Step 1: Compute the determinant of A using the formula ad − bc.

\det(A) = (4)(6) - (7)(2) = 24 - 14 = 10

Step 2: Since the determinant is 10 (not zero), the inverse exists. Swap the diagonal entries a and d, and negate the off-diagonal entries b and c.

\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix}

Step 3: Multiply the resulting matrix by 1 over the determinant.

A^{-1} = \frac{1}{10}\begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix}

Step 4: Verify by checking that AA⁻¹ equals the identity matrix.

AA^{-1} = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}\begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I \checkmark

Answer:

A^{-1} = \frac{1}{10}\begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix}

Another Example

This example uses a 3×3 matrix and the Gauss-Jordan (augmented matrix) method, which works for any size square matrix — unlike the 2×2 shortcut formula used in the first example.

Problem: Find the inverse of the 3×3 matrix using the augmented matrix (Gauss-Jordan) method: A = [[1, 0, 1], [0, 2, 1], [1, 1, 1]].

Step 1: Set up the augmented matrix [A | I] by placing the 3×3 identity matrix to the right of A.

\left[\begin{array}{ccc|ccc} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 2 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array}\right]

Step 2: Eliminate the entry in row 3, column 1 by replacing R₃ with R₃ − R₁.

\left[\begin{array}{ccc|ccc} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 2 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & -1 & 0 & 1 \end{array}\right]

Step 3: Swap R₂ and R₃ for convenience, then eliminate to get a leading 1 in row 2. After swapping: replace new R₃ with R₃ − 2·R₂.

\left[\begin{array}{ccc|ccc} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 1 \\ 0 & 0 & 1 & 2 & 1 & -2 \end{array}\right]

Step 4: Eliminate the entry in row 1, column 3 by replacing R₁ with R₁ − R₃.

\left[\begin{array}{ccc|ccc} 1 & 0 & 0 & -1 & -1 & 2 \\ 0 & 1 & 0 & -1 & 0 & 1 \\ 0 & 0 & 1 & 2 & 1 & -2 \end{array}\right]

Step 5: The left side is now the identity matrix, so the right side is A⁻¹. Verify: det(A) = 1(2·1 − 1·1) − 0 + 1(0·1 − 2·1) = 1 − 2 = −1 ≠ 0, confirming invertibility.

A^{-1} = \begin{bmatrix} -1 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & -2 \end{bmatrix}

Answer:

A^{-1} = \begin{bmatrix} -1 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & -2 \end{bmatrix}

Frequently Asked Questions

How do you know if a matrix has an inverse?

A matrix has an inverse if and only if it is square and its determinant is nonzero. If the determinant equals zero, the matrix is called singular and has no inverse. You can also check whether the row-reduced form of the matrix is the identity matrix; if it is, the matrix is invertible.

What is the difference between the inverse and the transpose of a matrix?

The inverse

A^{-1}

undoes multiplication by

A

, meaning

AA^{-1} = I

. The transpose

A^T

simply flips the matrix over its main diagonal, swapping rows and columns. Every matrix has a transpose, but only square matrices with nonzero determinants have inverses. These are completely different operations.

Why can't non-square matrices have inverses?

A true (two-sided) inverse requires

AA^{-1} = A^{-1}A = I

, which demands that both products be defined and result in identity matrices. This is only possible when

A

is square. Non-square matrices can have one-sided inverses (left or right), but not a full inverse.

Inverse of a Matrix vs. Transpose of a Matrix

	Inverse of a Matrix	Transpose of a Matrix
Definition	Matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I	Matrix Aᵀ formed by swapping rows and columns of A
Existence	Only for square matrices with nonzero determinant	Exists for every matrix of any size
Effect on original	"Undoes" multiplication by A	Reflects entries across the main diagonal
Key property	AA⁻¹ = I	(Aᵀ)ᵀ = A

Why It Matters

Matrix inverses are essential for solving systems of linear equations: if

A\mathbf{x} = \mathbf{b}

, you can find the solution directly as

\mathbf{x} = A^{-1}\mathbf{b}

. You will encounter them in linear algebra, physics, computer graphics, and data science whenever transformations need to be reversed. In precalculus and algebra courses, finding a 2×2 or 3×3 inverse is a standard exam topic.

Common Mistakes

Mistake: Forgetting to check that the determinant is nonzero before applying the inverse formula.

Correction: Always compute det(A) first. If det(A) = 0, the matrix is singular and has no inverse — attempting the formula will produce division by zero.

Mistake: In the 2×2 formula, swapping the wrong entries — negating the diagonal instead of the off-diagonal.

Correction: Remember: swap a and d (the main diagonal entries), and negate b and c (the off-diagonal entries). A common mnemonic is "swap the diagonal, flip the signs of the other two."

Related Terms

Identity Matrix — The result of multiplying a matrix by its inverse
Determinant — Must be nonzero for the inverse to exist
Square Matrix — Only square matrices can have inverses
Singular Matrix — A square matrix with no inverse (det = 0)
Invertible Matrix — A square matrix that does have an inverse
Gauss-Jordan Elimination — Row reduction method used to compute inverses
Matrix Multiplication — Operation used to verify AA⁻¹ = I
Classical Adjoint — Used in the adjoint method for finding inverses