Matrix Calculator

A matrix is a grid of numbers with rows and columns. A 2 × 3 matrix has 2 rows and 3 columns. Matrices hold systems of equations, rotations and other transformations, networks and data tables. This calculator does every common operation exactly: decimals you type become fractions (0.1 is 1/10), so there is no rounding until you ask for decimals at the end.

The formula

A ± B: add or subtract matching cells (same size)
A × B: cᵢⱼ = row i of A · column j of B (columns of A = rows of B)
2 × 2 determinant: det = ad − bc
n × n determinant: det(A) = Σ (−1)^(i + j) × aᵢⱼ × det(minor ij) along any row or column
inverse: A⁻¹ = adj(A) ÷ det(A), or [A | I] → [I | A⁻¹]
trace = a₁₁ + a₂₂ + … + aₙₙ; rank = number of pivots in the RREF
eigenvalues: det(A − λI) = 0; eigenvectors: (A − λI)v = 0
system Ax = b: row-reduce [A | b], or xᵢ = det(Aᵢ) ÷ det(A) (Cramer)

Worked example

Take A with rows (2, −1, 0), (1, 3, 2) and (0, 1, 4). Determinant by cofactors along row 1: 2 × (3 × 4 − 2 × 1) − (−1) × (1 × 4 − 2 × 0) + 0 × (1 × 1 − 3 × 0) = 2 × 10 + 4 + 0 = 24. By row reduction: R2 → R2 − (1/2)R1 makes row 2 (0, 7/2, 2), then R3 → R3 − (2/7)R2 makes row 3 (0, 0, 24/7). There were no swaps, so det(A) = 2 × 7/2 × 24/7 = 24.

Inverse: the cofactors are C₁₁ = 10, C₁₂ = −4, C₁₃ = 1, C₂₁ = 4, C₂₂ = 8, C₂₃ = −2, C₃₁ = −2, C₃₂ = −4, C₃₃ = 7. Transposing gives adj(A) with rows (10, 4, −2), (−4, 8, −4), (1, −2, 7). Divide by 24: A⁻¹ has rows (5/12, 1/6, −1/12), (−1/6, 1/3, −1/6) and (1/24, −1/12, 7/24).

Solving Ax = b with b = (1, 8, 9): x = A⁻¹b. x₁ = 5/12 × 1 + 1/6 × 8 − 1/12 × 9 = (5 + 16 − 9)/12 = 1. x₂ = −1/6 + 8/3 − 9/6 = (−1 + 16 − 9)/6 = 1. x₃ = 1/24 − 8/12 + 63/24 = (1 − 16 + 63)/24 = 2. So x₁ = 1, x₂ = 1, x₃ = 2, and A × (1, 1, 2) = (1, 8, 9) checks out.

Which sizes work for each operation

Row operations

There are three: swap two rows (Rᵢ ↔ Rⱼ), multiply or divide a row by a non-zero number (Rᵢ → Rᵢ ÷ 2), and add a multiple of one row to another (R₂ → R₂ − 3R₁). They never change the solutions of a system or the rank. For a determinant, a swap flips the sign, dividing a row by k divides the determinant by k, and adding a multiple of a row changes nothing.

Systems with no solution or many solutions

If a row of the reduced [A | b] reads 0 = 5, the equations contradict each other and there is no solution. The calculator then also gives the least-squares best fit, the x that makes Ax as close to b as possible (it solves AᵀA x = Aᵀb when that has one answer). If a column of A has no pivot, its unknown is free: it can be any number, written t, s, r …, and the others are given in terms of it.

Eigenvalues: exact or numeric

The eigenvalues are the roots of the characteristic polynomial det(λI − A). The calculator finds it exactly and tests fractions p/q (the rational root test). What is left, if it is a quadratic or splits into quadratics, is solved exactly with the quadratic formula, giving answers like 1/2 ± √5/2 or 1 ± 2i. Anything else is found numerically with the QR algorithm (repeated QR factorisations of the matrix) and shown as decimals. A repeated eigenvalue with fewer eigenvectors than its count means the matrix can’t be diagonalised.

Typing and pasting

• Copy cells in Excel, Google Sheets or Numbers and paste into any cell: the grid grows to fit (up to 6 × 6).
• Or open “Type or paste the whole matrix” and type rows separated by semicolons or new lines: 1 2 3; 4 5 6. [[1, 2], [3, 4]] works too.
• Use ↑ ↓ or Enter to move between rows. An empty cell counts as 0.
• “Copy answer” copies the result with tabs, so it pastes back into a spreadsheet as cells.

All results except numeric eigenvalues are exact. Very large powers can make numbers with hundreds of digits; the calculator stops if they get too big to show.