Inverse Matrix Calculator: 2×2, 3×3 & 4×4 with Steps

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Formula at a glance

Inverse Matrix Calculator: Adjugate method — A⁻¹ = adj(A) / det(A). Example: 3 × 3 matrix → A⁻¹: [0, 0, 1; −2, 1, 3; 3, −1, −5] — How the Inverse Matrix Calculator works: Adjugate method — A⁻¹ = adj(A) / det(A)

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📘 About the Inverse Matrix Calculator

The inverse of a square matrix A is the matrix A⁻¹ with A × A⁻¹ = I, the identity. It exists only when the determinant is not zero. Inverses solve systems of equations (X = A⁻¹B), undo transformations and appear throughout statistics and engineering.

Enter a square matrix up to 6 × 6. The calculator finds the determinant, then (up to 4 × 4) the cofactor matrix and the adjugate, divides by the determinant, and confirms the result by Gauss–Jordan elimination on [A | I]. Fractions stay exact, and decimals are shown too.

🧮 Formula

Adjugate method

A⁻¹ = adj(A) / det(A)

2 × 2

[a b; c d]⁻¹ = (1/(ad − bc))·[d −b; −c a]

Gauss–Jordan

[A | I] → [I | A⁻¹]

✏️ Worked examples

3 × 3 matrix

You enter
Matrix A	2 1 1; 1 3 2; 1 0 0

You get
A⁻¹	[0, 0, 1; −2, 1, 3; 3, −1, −5]
det(A)	−1
A	[2, 1, 1; 1, 3, 2; 1, 0, 0]

[0, 0, 1; −2, 1, 3; 3, −1, −5].

2 × 2 matrix

You enter
Matrix A	4 7; 2 6

You get
A⁻¹	[3/5, −7/10; −1/5, 2/5]
det(A)	10
A	[4, 7; 2, 6]
A⁻¹ in decimals	[0.6, −0.7; −0.2, 0.4]

[3/5, −7/10; −1/5, 2/5].

🧭 How to use this calculator

What you enter

Matrix ARows separated by ; or new lines, entries by spaces or commas. Fractions like 1/2 are fine.

What you get

A⁻¹
det(A)
A

❓ Frequently asked questions

When does a matrix have no inverse?

When its determinant is 0 (a singular matrix) — its rows are linearly dependent.

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

Swap a and d, change the signs of b and c, and divide by ad − bc.

Why use fractions instead of decimals?

Exact fractions avoid rounding errors, so A × A⁻¹ is exactly the identity.

A⁻¹ Inverse Matrix Calculator (2×2, 3×3, 4×4) with Steps