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det A · adj A ÷ det A

Matrix calculator

Enter a 2×2 or 3×3 matrix to get its determinant, trace and inverse, with the identity check computed so the inverse can be verified.

Short answer: [[4, 7], [2, 6]] has determinant 10 and inverse [[0.6, −0.7], [−0.2, 0.4]]; multiplying them returns the identity. The matrix [[1, 2], [2, 4]] has determinant 0 and therefore no inverse at all.

What the determinant tells you

It is the factor by which the matrix scales area, and its sign says whether orientation flips. Zero means the transformation collapses a dimension: every point lands on a line or a plane, and no inverse can bring them back. That is exactly the case this page reports instead of dividing.

Worked example

Solving 2x + 3y = 8, x + 4y = 11: the coefficient matrix [[2, 3], [1, 4]] has determinant 5, and its inverse times the constants gives x = −0.2 and y = 2.8. The identity check is the reason you can trust the inverse before using it.

Matrix questions

What is a determinant?

A single number that says whether the matrix can be inverted and how it scales area or volume. A 2×2 determinant is ad − bc; a 3×3 expands along the first row into three 2×2 determinants.

When does a matrix have no inverse?

Exactly when its determinant is zero — the rows are linearly dependent, so the transformation flattens space and cannot be undone. The page says that rather than dividing by zero and printing infinities.

How is the inverse found?

A⁻¹ = adjugate(A) ÷ det(A), where the adjugate is the transpose of the cofactor matrix. The page computes A × A⁻¹ as well, so you can see it comes out as the identity rather than taking the formula’s word for it.

Why only 2×2 and 3×3?

Because those are the sizes that can be shown with their working, and larger systems belong in a proper linear-algebra tool. A 4×4 determinant by hand is 24 products and an easy place to make a mistake.

Related tools

Linear equation calculatorOne-variable equations with the working shown.GCD and LCM calculatorNumber theory for the same determinant steps.Unit converterCross-check dimensions and units.