Inverse of a Square Matrix: Formulas, Method, Examples

The inverse of a square matrix A, written A⁻¹, is the matrix that "undoes" A. When you multiply a matrix by its inverse, you get the identity matrix: A × A⁻¹ = A⁻¹ × A = I. It plays the same role for matrices that the reciprocal 1/a plays for ordinary numbers.

The inverse matters because it lets us solve matrix equations the way we solve ordinary equations. Instead of guessing, we can divide both sides by the coefficient matrix. Inverses also reverse geometric transformations, which is why graphics software, robotics and cryptography all depend on them.

Key Concepts

• Identity matrix I: the square matrix with 1s on the main diagonal and 0s elsewhere; multiplying any matrix by I leaves it unchanged.
• Inverse A⁻¹: the matrix satisfying A × A⁻¹ = A⁻¹ × A = I.
• Singular matrix: a square matrix with det(A) = 0; it has no inverse.
• Non-singular matrix: a square matrix with det(A) ≠ 0; its inverse exists and is unique.
• Only square matrices can have a two-sided inverse.

Formulas

• Adjoint formula: A⁻¹ = (1 / det(A)) × adj(A). Assumption: A is square and det(A) ≠ 0.
• Product rule: (AB)⁻¹ = B⁻¹A⁻¹. Note the reversed order, just like taking off socks before shoes.
• Double inverse: (A⁻¹)⁻¹ = A.
• Transpose rule: (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
• Determinant rule: det(A⁻¹) = 1 / det(A).
• Scalar multiple: (kA)⁻¹ = (1/k) × A⁻¹ for any non-zero number k.

Worked Example

Find the inverse of A = [[2, 1], [3, 4]].

1. Compute the determinant: det(A) = 2 × 4 − 1 × 3 = 8 − 3 = 5. Since 5 ≠ 0, the inverse exists.
2. Find the adjoint. The cofactor matrix is [[4, −3], [−1, 2]], so adj(A) = [[4, −1], [−3, 2]].
3. Divide by the determinant: A⁻¹ = (1/5) × [[4, −1], [−3, 2]] = [[4/5, −1/5], [−3/5, 2/5]].
4. Verify by multiplication: A × A⁻¹ = [[2, 1], [3, 4]] × [[4/5, −1/5], [−3/5, 2/5]].
5. Row 1: 2(4/5) + 1(−3/5) = (8 − 3)/5 = 1; 2(−1/5) + 1(2/5) = 0. Row 2: 3(4/5) + 4(−3/5) = 0; 3(−1/5) + 4(2/5) = (−3 + 8)/5 = 1. The product is [[1, 0], [0, 1]] = I, so the inverse is correct.

Common Misconceptions

• The inverse is not found by dividing 1 by each entry. The formula uses the adjoint and the determinant of the whole matrix, never entry-by-entry reciprocals.
• Not every square matrix has an inverse. Singular matrices with zero determinant cannot be inverted.
• Order matters in (AB)⁻¹ = B⁻¹A⁻¹. Writing A⁻¹B⁻¹ is wrong because matrix multiplication does not commute.

Real-World Applications

• Solving linear equations: the system AX = B is solved by X = A⁻¹B.
• Computer graphics: undoing rotations, scalings and translations applied to 3D objects.
• Cryptography: the Hill cipher encrypts with a matrix and decrypts with its inverse.
• Economics: input-output models invert matrices to find required production levels.

The inverse of a square matrix exists exactly when its determinant is non-zero, and it is computed as A⁻¹ = adj(A) / det(A). Always check your answer by multiplying back to the identity matrix, and remember that singular matrices simply have no inverse.