Matrix Inversion Method Calculator

Enter the coefficients of a 2×2 or 3×3 system of linear equations and solve using matrix inversion. The calculator shows the cofactor matrix, adjugate, inverse A⁻¹, and the final product X = A⁻¹·B step by step.

Number of variables:(matrix inversion requires computing cofactors; limited to 3×3)

x₁	x₂	x₃	= b

Notes

How Matrix Inversion Works

For the system Ax = b, if A is invertible (det(A) ≠ 0), the solution is x = A⁻¹b. The inverse is computed using the adjugate: A⁻¹ = adj(A) / det(A).

Compute det(A). If det(A) = 0, the matrix is singular and has no inverse.
Compute the cofactor matrix C, where Cᵢⱼ = (−1)ⁱ⁺ʲ · Mᵢⱼ.
Transpose C to get the adjugate: adj(A) = Cᵀ.
Divide by the determinant: A⁻¹ = adj(A) / det(A).
Multiply: X = A⁻¹ · B to get the solution vector.

⚠Matrix inversion is limited to square matrices with det(A) ≠ 0. This calculator supports 2×2 and 3×3 systems only, since cofactor expansion for larger matrices becomes impractical.

Worked Example — 2×2

System: 3x₁ + x₂ = 7, x₁ + 2x₂ = 4

A = (3112), det (A) = 6 - 1 = 5

A^{- 1} = \frac{1}{5} (2 - 1 - 1 3), X = A^{- 1} B = \frac{1}{5} (2 (7) + (- 1) (4) (- 1) (7) + 3 (4)) = (21)

Worked Example — 3×3

System: x₁ + 2x₂ + 3x₃ = 14, x₁ + x₂ + x₃ = 6, 2x₁ + x₂ + x₃ = 8

A = 112211311, det (A) = 1 (0) + (- 2) (- 1) + 3 (- 1) = 0 + 2 - 3 = - 1

Compute cofactors, transpose to get adj(A), divide by det(A) = −1 to get A⁻¹, then X = A⁻¹·B:

A^{- 1} = \frac{1}{- 1} 01 - 1 1 - 5 3 - 1 2 - 1^{⊤} = 0 - 1 1 - 1 5 - 3 1 - 2 1

X = A^{- 1} B = 0 (14) + (- 1) (6) + 1 (8) (- 1) (14) + 5 (6) + (- 2) (8) 1 (14) + (- 3) (6) + 1 (8) = 204

ℹSolution: x₁ = 2, x₂ = 0, x₃ = 4. Verify: 2+0+12 = 14 ✓, 2+0+4 = 6 ✓, 4+0+4 = 8 ✓

Matrix Inversion Method — Full Notes — Theory, cofactors, adjugate, and worked examples
Matrix Inversion Formula Reference — Full formula reference with variable definitions

Frequently Asked Questions

What is the matrix inversion method?

The matrix inversion method solves the system Ax = b by computing the inverse of A and multiplying: x = A⁻¹b. The inverse exists only when det(A) ≠ 0.

What is the adjugate matrix?

The adjugate (or adjoint) of A is the transpose of the cofactor matrix: adj(A) = Cᵀ. It is used to compute the inverse: A⁻¹ = adj(A) / det(A).

Why is matrix inversion limited to 3×3 here?

Computing cofactors requires calculating (n−1)×(n−1) determinants for every entry. For 4×4 matrices this means 16 determinants of 3×3 matrices, becoming unwieldy. Gaussian elimination is more practical for larger systems.

What happens if det(A) = 0?

If det(A) = 0, the matrix A is singular and has no inverse. The system either has no solution (inconsistent) or infinitely many solutions (dependent). Use Gaussian elimination to determine which case applies.