Matrix Calculator

Find the determinant, inverse and transpose of a matrix, and add or multiply matrices, with exact fractions.

Matrix A	2 1 0 1 3 1 0 1 4
Operation	Determinant of A (with cofactor expansion)

Matrix A

One row per line, numbers separated by spaces. Fractions like 1/2 are fine.

Operation

det(A)

det(A) = 18. A is invertible.

Exact fraction. Checked exactly.

Method: Cofactor expansion along row 1
Worked out with: AllCalcHub exact fractions
Check: Every step recomputed with exact fractions

A

	col 1	col 2	col 3
row 1	2	1	0
row 2	1	3	1
row 3	0	1	4

1
Minor M₁1
M₁1 = 3 × 4 − 1 × 1 = 11
2
Minor M₁2
M₁2 = 1 × 4 − 1 × 0 = 4
3
Minor M₁3
M₁3 = 1 × 1 − 3 × 0 = 1
4
Expand along row 1: |A| = a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃
2(11) − 1(4) + 0(1) = 18

About the Matrix Calculator

Matrices are grids of numbers used to solve systems of equations, transform shapes in graphics and model networks. This calculator works with exact fractions, so an inverse like 1/3 stays exact instead of 0.333333.

Type one row per line. A matrix has an inverse only if its determinant isn't zero.

The formula

det [[a, b], [c, d]] = ad − bc; A⁻¹ = (1/det A) [[d, −b], [−c, a]]

A × B: entry (i, j) = row i of A · column j of B
Larger matrices: row reduction (Gaussian elimination)

Worked example

A = [[2, 1], [5, 3]]

det = 2·3 − 1·5 = 1
A⁻¹ = [[3, −1], [−5, 2]]

Questions people ask

Why can't I multiply these matrices?

The number of columns in A must equal the number of rows in B.

Is A × B the same as B × A?

Usually not; matrix multiplication isn't commutative.