Matrix Calculator

Enter one or two matrices and choose an operation. The calculator works with exact fractions, so inverses and solutions show 1/3 rather than 0.3333, and it lists the row operations used for Gauss–Jordan elimination so you can follow or check your own working. Matrices up to 8 × 8 are supported.

What the calculator can do

Entering matrices

Type each row on its own line and separate entries with spaces or commas. You can use integers, decimals (0.25) or fractions (3/4); decimals are converted to exact fractions. For a system of equations, put the coefficients in A and the right-hand side in B as a single column. The default example is the system 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3, whose solution is x = 2, y = 3, z = −1.

Gauss–Jordan elimination

Most operations here use Gauss–Jordan elimination: three row operations — swapping rows, multiplying a row by a non-zero number, and adding a multiple of one row to another — turn the matrix into reduced row echelon form, with leading 1s and zeros above and below them. Applied to [A | b], it solves linear systems; applied to [A | I], it produces the inverse; and the product of the pivots (with sign changes for swaps) gives the determinant. The steps list shows the operations used, such as “R2 − (−3/2)·R1”.

Determinants and invertibility

A square matrix is invertible exactly when its determinant is not zero. The determinant also tells you how the matrix scales area or volume: a 2 × 2 matrix with determinant 3 triples areas; a negative determinant flips orientation. For a 2 × 2 matrix [[a, b], [c, d]], det = ad − bc and the inverse is (1/det)·[[d, −b], [−c, a]].

Matrix multiplication rules

• The product AB exists only when A’s column count equals B’s row count; the result has A’s rows and B’s columns.
• Order matters: AB is usually not equal to BA.
• Each entry of AB is the dot product of a row of A with a column of B.
• The identity matrix I works like 1: AI = IA = A.
• (AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹.

Worked example

For A = [[2, 1, −1], [−3, −1, 2], [−2, 1, 2]] and b = [8, −11, −3], “Solve Ax = b” reduces the augmented matrix to reduced row echelon form and reads off x₁ = 2, x₂ = 3, x₃ = −1. Switching to “Determinant” gives det(A) = −1, confirming the system has a unique solution, and “Inverse” shows A⁻¹ with integer entries because the determinant is ±1.

Where matrices are used

Matrices appear throughout science and engineering: solving circuits and structural systems, transforming coordinates in computer graphics, representing networks, running statistics and machine learning, and modelling population or economic changes over time with matrix powers.

Special matrices

Eigenvalues and beyond

Many courses move on from determinants and inverses to eigenvalues and eigenvectors, which describe directions a matrix only stretches. For a 2 × 2 matrix, eigenvalues solve λ² − (trace)·λ + det = 0. The determinant equals the product of the eigenvalues and the trace (sum of the diagonal) equals their sum — a handy check once you study them.

Checking your answers

• Multiply A by the inverse you found — you should get the identity matrix.
• Substitute a solution back into each equation.
• For a triangular matrix, the determinant is simply the product of the diagonal.
• Swapping two rows changes the sign of the determinant; multiplying a row by k multiplies it by k.
• Rank never exceeds the smaller of the number of rows and columns.

Limits and accuracy

The calculator uses exact rational arithmetic with standard JavaScript numbers, which is precise for typical classroom and textbook matrices. Very large entries or large matrices can overflow the safe integer range; in that case results may lose accuracy. For numerical work with large matrices, specialist software using floating-point methods is more appropriate.

Privacy

All calculations run in your browser; nothing is uploaded or stored.