Matrix Calculator
This matrix calculator finds the determinant, inverse, transpose, rank, trace, sum, product and power of matrices up to 8×8. Results are exact fractions, with the Gaussian elimination steps shown.
Source: Wolfram MathWorld – Gaussian elimination. Updated: .
How it is calculated
A matrix is a rectangular grid of numbers; it describes linear systems of equations, rotations and much more. The tool works with exact fractions, so there are no rounding errors – for example in the inverse.
What it calculates
- Determinant: it shows whether a square matrix can be inverted (det ≠ 0). It is computed by Gaussian elimination; each row swap flips the sign.
- Inverse: the matrix A⁻¹ with A · A⁻¹ = I, found by Gauss-Jordan elimination and checked automatically.
- Rank: the number of linearly independent rows, read from the echelon form.
- Product: entry (i, j) is row i of A times column j of B, so A needs as many columns as B has rows. Unlike numbers, A · B ≠ B · A in general.
- Power: Aⁿ by repeated multiplication; A⁰ is the identity matrix.
Input
One text line per matrix row, values separated by spaces. A semicolon also works as a row break: “1 2; 3 4”. Whole numbers, decimals with a comma or a point and fractions like 3/4 are accepted.
To solve a system of equations use the system of equations calculator; the fraction calculator reduces fractions, and the calculator helps with single steps.
RREF, LU decomposition and eigenvalues
- Reduced row echelon form (RREF): Gauss-Jordan elimination with every row operation listed. “1 2 3; 4 5 6; 7 8 9” becomes “1 0 -1; 0 1 2; 0 0 0”, so the rank is 2.
- LU decomposition: P · A = L · U with a lower triangular L (ones on the diagonal) and an upper triangular U. Rows are only swapped when a pivot is 0.
- Eigenvalues: rational roots of the characteristic polynomial det(λI − A) are found exactly, a remaining quadratic factor exactly with a square root. For “2 1; 1 2” you get λ = 1 and λ = 3 with eigenvectors (-1; 1) and (1; 1). Whatever cannot be solved exactly is approximated and marked with ≈.
- Adjugate: the transpose of the cofactor matrix; A · adj(A) = det(A) · I.
You can paste a matrix straight from a spreadsheet (tabs between cells). Notations such as [[1, 2], [3, 4]] or [1 2; 3 4] are read as well.
Frequently asked questions
How do I find the determinant of a 3×3 matrix?
Reduce it to triangular form by Gaussian elimination and multiply the diagonal, or use the rule of Sarrus. The tool shows the row operations step by step.
When does a matrix have no inverse?
When its determinant is 0. Then the rows depend on each other (rank below the number of rows) and the system of equations has no unique solution.
How do you multiply two matrices?
Each entry of the product is the sum of products of a row of A and a column of B. The number of columns of A must equal the number of rows of B.
What does the rank of a matrix mean?
It tells how many rows (or columns) are linearly independent – how many rows stay non-zero after Gaussian elimination.
Why does the tool show fractions instead of decimals?
Fractions are exact: 1/3 stays 1/3 instead of 0.333…. If you prefer decimals, switch the display.
How do I find the eigenvalues of a matrix?
The eigenvalues are the roots of the characteristic polynomial det(λI − A). For “2 1; 1 2” it is λ² − 4λ + 3, so the eigenvalues are 1 and 3. The calculator shows the polynomial and lists the eigenvectors.
Sources and legal basis
- Wolfram MathWorld – Gaussian elimination
- Wolfram MathWorld – Determinant
- Wolfram MathWorld – Matrix inverse
- Wolfram MathWorld – Eigenvalue
- Wolfram MathWorld – LU decomposition
As of: · Editorial policy
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