What is the inverse of a matrix?

The inverse of a square matrix A, written A⁻¹, is the matrix that when multiplied by A gives the identity matrix: A · A⁻¹ = A⁻¹ · A = I. The identity matrix has 1s on the main diagonal and 0s everywhere else. The matrix inverse is the matrix equivalent of dividing by a number — just as multiplying by 1/x undoes multiplication by x, multiplying by A⁻¹ undoes multiplication by A. Only square matrices can have inverses, and not all square matrices are invertible.

When does a matrix not have an inverse?

A matrix has no inverse when its determinant equals zero. Such a matrix is called singular or degenerate. Geometrically, a singular matrix maps all of space to a lower-dimensional subspace — for example, it might collapse a plane to a line — so the transformation cannot be undone. Practically, a singular matrix in a system of equations means the system either has no solution or infinitely many solutions. A 2×2 matrix [[a,b],[c,d]] is singular when ad − bc = 0, meaning its rows (or columns) are linearly dependent.

What is the adjugate matrix?

The adjugate (also called the adjoint) of a matrix is the transpose of its cofactor matrix. Each entry in the cofactor matrix is the signed minor: the determinant of the submatrix obtained by removing that row and column, multiplied by (−1)^(i+j). Transposing this gives the adjugate. The inverse is then A⁻¹ = (1/det) · adj(A). For a 2×2 matrix [[a,b],[c,d]], the adjugate is simply [[d,−b],[−c,a]].

How do you find the inverse of a 2×2 matrix?

For a 2×2 matrix A = [[a,b],[c,d]]: (1) compute the determinant: det = ad − bc; (2) if det = 0, the matrix is not invertible; (3) form the adjugate by swapping a and d, and negating b and c: [[d,−b],[−c,a]]; (4) divide every entry by the determinant: A⁻¹ = (1/det) · [[d,−b],[−c,a]]. Example: A = [[2,1],[5,3]], det = 6−5 = 1, A⁻¹ = [[3,−1],[−5,2]].

What is the cofactor matrix?

The cofactor matrix C of a matrix A has entry C_ij = (−1)^(i+j) · M_ij, where M_ij is the minor — the determinant of the submatrix obtained by deleting row i and column j from A. The sign pattern for a 3×3 matrix is [[+,−,+],[−,+,−],[+,−,+]]. For a 3×3 matrix, you compute nine 2×2 determinants, apply the sign pattern, and arrange the results. The adjugate is the transpose of the cofactor matrix.

Why is the matrix inverse useful?

The matrix inverse is central to linear algebra and its applications. In computer graphics, inverse matrices undo transformations (rotations, scaling, reflections) to convert from screen coordinates back to world coordinates. In engineering and physics, inverse matrices solve systems of linear equations: if Ax = b, then x = A⁻¹b. In statistics, the inverse of the covariance matrix appears in multivariate analysis and regression. Cryptography uses matrix inverses for certain encryption and decryption operations.

How is matrix inversion used to solve linear equations?

If you have a system Ax = b (where A is an n×n matrix of coefficients, x is the unknown vector, and b is the right-hand side), you can solve it as x = A⁻¹b — just multiply both sides by the inverse. For example, the 2×2 system 2x + y = 5, 5x + 3y = 13 becomes A = [[2,1],[5,3]], b = [5,13], A⁻¹ = [[3,−1],[−5,2]], and x = A⁻¹b = [3×5 + (−1)×13, (−5)×5 + 2×13] = [2, 1]. For large systems, Gaussian elimination is faster than computing A⁻¹ directly.

What does the matrix inverse represent geometrically?

A matrix A represents a linear transformation (a combination of rotation, scaling, and shearing of space). The inverse A⁻¹ represents the exact opposite transformation that undoes A. If A rotates the plane by 30° clockwise, then A⁻¹ rotates it 30° counterclockwise. If A stretches the x-axis by a factor of 3, then A⁻¹ compresses it by 1/3. A matrix with det(A) = 0 is non-invertible because its transformation collapses space to a lower dimension — an operation that cannot be reversed.

Inverse Matrix Calculator

Finds the inverse of any 2×2 or 3×3 matrix — shows the determinant, cofactor matrix, and adjugate with step-by-step work.

How to Use the Inverse Matrix Calculator

This inverse matrix calculator supports 2×2 and 3×3 matrices — select the size, enter all entries, then click Find Inverse to compute the inverse matrix, determinant, and (for 3×3 matrices) the cofactor matrix and adjugate. Toggle Show step-by-step work to see the full derivation. If the determinant is zero, the matrix is singular and has no inverse — the calculator will show an error explaining why. For solving systems of linear equations by row reduction, see our RREF calculator.

Results are shown as decimals rounded to four decimal places. The step panel shows the intermediate cofactor values and the adjugate before dividing by the determinant.

How to Find the Inverse of a 2×2 Matrix

For a 2×2 matrix A = [[a, b], [c, d]], the inverse has a simple closed-form formula. There are just four steps:

Compute the determinant: det(A) = a·d − b·c
Check invertibility: if det = 0, stop — the matrix is singular and has no inverse
Form the adjugate: swap the diagonal entries (a and d), and negate the off-diagonal entries (b and c) → [[d, −b], [−c, a]]
Divide by the determinant: A⁻¹ = (1/det) · [[d, −b], [−c, a]]

Example: A = [[4, 7], [2, 6]]. det = 4·6 − 7·2 = 24 − 14 = 10. A⁻¹ = (1/10) · [[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]]. Verification: A · A⁻¹ = [[4·0.6 + 7·(−0.2), …], …] = [[1, 0], [0, 1]].

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How to Find the Inverse of a 3×3 Matrix

For a 3×3 matrix, the adjugate method has three main stages: compute the determinant, build the cofactor matrix, transpose it to get the adjugate, then divide by the determinant.

Step 1 — Compute the 3×3 Determinant

Expand along the first row using the rule of Sarrus or cofactor expansion:

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

where [[a, b, c], [d, e, f], [g, h, i]] are the matrix entries. If det = 0, the matrix is not invertible. For determinant-only calculations, our determinant calculator handles larger matrices.

Step 2 — Compute the Cofactor Matrix

Each cofactor C_ij = (−1)^(i+j) × M_ij, where M_ij is the 2×2 minor determinant obtained by deleting row i and column j. The sign pattern is:

+ − +
− + −
+ − +

Compute all nine 2×2 determinants, apply the sign, and arrange into the cofactor matrix C.

Step 3 — Transpose to Get the Adjugate

The adjugate adj(A) is the transpose of C: swap entry C_ij with C_ji. The adjugate has the property that A · adj(A) = det(A) · I, which is why dividing by the determinant gives the inverse.

Step 4 — Divide by the Determinant

A⁻¹ = (1 / det(A)) · adj(A). Divide every entry of the adjugate by the determinant to obtain the inverse matrix.

Verification: A · A⁻¹ = I

Always verify your result by multiplying the original matrix by its inverse — the product must equal the identity matrix (1s on the diagonal, 0s elsewhere). Small floating-point errors of ±0.0001 are normal and expected when working with decimals. If any off-diagonal entry is larger than about 0.001, recheck your inputs or the computation.

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When a Matrix Is Not Invertible

A matrix is singular (not invertible) when its determinant is zero. This happens when:

Two rows (or columns) are identical — for example, [[1, 2], [1, 2]]
One row is a multiple of another — for example, [[1, 2], [2, 4]] has det = 4 − 4 = 0
One row is a zero row — [[1, 2], [0, 0]] has det = 0
Rows are linearly dependent — in 3×3 matrices, if one row can be expressed as a linear combination of the other two

In a system of equations, a singular coefficient matrix means the system either has no solution (inconsistent) or has infinitely many solutions. Use the RREF calculator to distinguish between the two cases.

Applications of the Matrix Inverse

Solving linear systems: if Ax = b, then x = A⁻¹b — but for large systems, row reduction is more efficient in practice
Computer graphics: inverse transformation matrices convert from screen space back to world space, enabling ray tracing and picking
Statistics: the inverse covariance matrix (precision matrix) appears in multivariate normal distributions and linear discriminant analysis
Control systems: transfer functions and state-space models require matrix inverses for stability analysis
Cryptography: Hill cipher encryption and decryption use matrix multiplication and matrix inverses over modular arithmetic

Sources & References

Inverse of a Matrix — Khan Academy — Khan Academy
Inverse of a Matrix — Math is Fun — Math is Fun

Inverse Matrix Calculator