Inverse Matrix Calculator

The inverse of a square matrix A is the matrix A⁻¹ with A·A⁻¹ = A⁻¹·A = I, the identity matrix. For A = [[4, 7], [2, 6]], A⁻¹ = [[0.6, −0.7], [−0.2, 0.4]].

• A⁻¹ exists exactly when det(A) ≠ 0. Such a matrix is called invertible or non-singular.
• Only square matrices (n × n) can have an inverse.
• Geometrically, A moves points and A⁻¹ moves them back. A 2 × 2 matrix maps the unit square to a parallelogram of area |det A|.
• When det(A) = 0, A squashes the plane onto a line, and no matrix can undo that.

For A = [[a, b], [c, d]], swap a and d, change the signs of b and c, and divide by det = ad − bc.

For A = [[4, 7], [2, 6]]:

1. det = 4 × 6 − 7 × 2 = 24 − 14 = 10.
2. Swap and change signs: [[6, −7], [−2, 4]].
3. Divide by 10: A⁻¹ = [[0.6, −0.7], [−0.2, 0.4]].

Check one entry of A·A⁻¹: row 1 times column 1 is 4 × 0.6 + 7 × (−0.2) = 2.4 − 1.4 = 1.

Type the matrix one row per line, and the calculator returns A⁻¹ in exact fractions with every row operation. The steps:

1. Choose Any size for a matrix up to 8 × 8, 2 × 2 formula for 4 entries, or Solve Ax = b for a linear system.
2. Type each row on its own line, with numbers split by spaces or commas. [[1, 2], [3, 4]] and 1 2; 3 4 work too.
3. Read A⁻¹, the determinant, the check A·A⁻¹ = I and the condition number.
4. Open the [A | I] diagram or the 2 × 2 geometry view, and the Gauss–Jordan steps.

When det(A) = 0, the calculator says which column has no pivot instead of giving an answer.

To invert a 3 × 3 matrix, write [A | I], use row operations to turn the left half into I, and read A⁻¹ from the right half.

1. Write A next to the identity: [A | I].
2. In column 1, get a 1 on the diagonal (swap rows or divide), then subtract multiples of that row to make the rest of the column 0.
3. Repeat for columns 2 and 3.
4. When the left half is I, the right half is A⁻¹.

For A = [[2, 1, 1], [1, 3, 2], [1, 0, 0]], 9 row operations give A⁻¹ = [[0, 0, 1], [−2, 1, 3], [3, −1, −5]]. The pivots multiply to det(A) = −1, with one sign flip for the row swap.

3 allowed operations: swap 2 rows, multiply a row by a non-zero number, and add a multiple of one row to another.

The adjugate formula is A⁻¹ = adj(A) / det(A), where adj(A) is the transpose of the cofactor matrix.

1. Find det(A).
2. Replace each entry by its minor: the determinant left after deleting its row and column.
3. Apply the sign pattern + − + / − + − / + − +.
4. Transpose (swap rows and columns) to get adj(A).
5. Divide every entry by det(A).

The 2 × 2 formula is the adjugate method in miniature: adj([[a, b], [c, d]]) = [[d, −b], [−c, a]]. For 4 × 4 and larger, Gauss–Jordan takes far fewer steps.

A square matrix has no inverse when det(A) = 0: its rows (or columns) depend on each other, and A flattens space.

• [[1, 2], [2, 4]]: row 2 is 2 × row 1, so det = 4 − 4 = 0.
• Any matrix with a row or column of zeros is singular.
• Any matrix with 2 equal rows is singular.
• Non-square matrices have no inverse at all; they can have a one-sided or pseudo-inverse instead.

Near-singular matrices are risky in practice. As det(A) shrinks towards 0, the entries of A⁻¹ grow without bound, and small rounding errors in A swing the answer.

The inverse reverses products, commutes with the transpose, and inverts the determinant.

• (AB)⁻¹ = B⁻¹A⁻¹: undo the last step first, like taking off socks after shoes.
• (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
• det(A⁻¹) = 1/det(A).
• (A⁻¹)⁻¹ = A and (kA)⁻¹ = (1/k)A⁻¹ for k ≠ 0.
• Diagonal matrices: invert each diagonal entry. diag(2, 5)⁻¹ = diag(1/2, 1/5).
• Orthogonal matrices (rotations, reflections): A⁻¹ = Aᵀ.

In general A⁻¹B⁻¹ ≠ (AB)⁻¹, since matrix products depend on order.

A system Ax = b with det(A) ≠ 0 has exactly one solution, x = A⁻¹b.

For 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3:

1. A = [[2, 1, −1], [−3, −1, 2], [−2, 1, 2]] and b = (8, −11, −3).
2. A⁻¹ = [[4, 3, −1], [−2, −2, 1], [5, 4, −1]].
3. x = A⁻¹b = (4·8 + 3·(−11) + (−1)·(−3), …) = (2, 3, −1).

Solvers such as NumPy's linalg.solve skip forming A⁻¹ and eliminate on [A | b] directly; it is faster and more accurate. A⁻¹ pays off when the same A meets many different b vectors.

NumPy has np.linalg.inv(A), MATLAB has inv(A), Excel has MINVERSE and R has solve(A).

• Python: np.linalg.inv(np.array([[4, 7], [2, 6]])) returns [[0.6, −0.7], [−0.2, 0.4]] in floating point.
• SymPy: Matrix([[4, 7], [2, 6]]).inv() returns exact fractions.
• MATLAB: inv(A) or A^-1; A\b solves a system without forming the inverse.
• Excel and Google Sheets: =MINVERSE(A1:B2) spills the inverse into a 2 × 2 range; =MMULT(MINVERSE(A), b) solves a system.
• Wolfram Language: Inverse[{{4, 7}, {2, 6}}].

The table lists common 2 × 2 and 3 × 3 matrices with their exact inverses. Click a row to load the matrix into the calculator.

Patterns to spot: a diagonal matrix inverts entry by entry, a shear flips the sign of its off-diagonal entry, a rotation inverts to the opposite rotation, and an upper triangular matrix stays upper triangular.

Inverse matrices undo transformations, decode messages and solve the linear systems behind graphics, engineering and economics.

• Computer graphics: a camera or object transform is undone with its inverse. A rotation R(θ) has R(θ)⁻¹ = R(−θ) = R(θ)ᵀ.
• Hill cipher: the key [[3, 3], [2, 5]] has det = 9, and 9⁻¹ mod 26 = 3, so the decoding key is [[15, 17], [20, 9]] mod 26.
• Economics: the Leontief model finds production x = (I − A)⁻¹d for a demand d.
• Circuits and structures: node voltages and member forces come from solving Ax = b.
• Statistics: least squares uses (XᵀX)⁻¹Xᵀy for regression coefficients.