Matrix Inverse

Matrix Input

Matrix A 2 × 2

Examples

This tab auto-populates from the matrix you entered in Find A⁻¹. Switch back to change the matrix.

Matrix A (from Tab 1)

Result

Enter a 2×2 matrix above and press Calculate to find A⁻¹ and see the step-by-step work.

det(A) = —

Matrix is Singular — Inverse does not exist.
det(A) = 0 means A maps space to a lower dimension. No inverse formula applies.

A⁻¹

A · A⁻¹

The Inverse Formula

A⁻¹ = (1/det(A)) · adj(A)

For a 2×2 matrix A = [[a,b],[c,d]]:

1. Compute det(A) = ad − bc.
2. Form the adjugate: swap the diagonal entries (a ↔ d) and negate the off-diagonal entries (b → −b, c → −c).
3. Divide every entry by det(A).

Result: A⁻¹ = (1/det) · [[d, −b], [−c, a]]

Only works when det(A) ≠ 0. A matrix with zero determinant is called singular and has no inverse.

Always simplify each entry to a fraction — e.g., write 3/5 instead of 0.6 to keep exact values.

Why Inverses Matter

Ax = b → x = A⁻¹b

Solving matrix equations: A⁻¹ lets you "undo" matrix multiplication. If Ax = b, multiply both sides on the left by A⁻¹ to get x = A⁻¹b — just like dividing both sides of a scalar equation.

Invertible = nonsingular = full rank: An invertible 2×2 matrix has det ≠ 0, rank 2, and its two rows (or columns) are linearly independent.

A singular matrix maps the plane to a line or a point — information is lost. There is no way to "undo" that collapse, so no inverse exists.

A⁻¹ · A = A · A⁻¹ = I (the identity matrix)
(AB)⁻¹ = B⁻¹ · A⁻¹ (reverse order!)
(Aᵀ)⁻¹ = (A⁻¹)ᵀ — inverting commutes with transposing.
det(A⁻¹) = 1 / det(A)

Matrix inverses still tricky?