Work with one or two matrices up to 8 by 8 and get the determinant, inverse, RREF, rank, eigenvalues, LU and more. Answers stay in exact fractions like 3/7, so nothing is lost to rounding, and each step is listed so you can follow along or check your own work. It is free, with no ads and no sign-up.
How to use the matrix calculator
- Set the size of Matrix A and fill in the cells. Use Paste to bring in numbers from a spreadsheet, or Identity and Random for quick starts.
- For an operation on two matrices, fill in Matrix B as well.
- Pick an operation: Determinant, Inverse, RREF, Rank, Eigenvalues, LU, Transpose, Trace, k times A or A to the power n, or A plus B, A minus B and A times B.
- Read the answer and the steps. Copy LaTeX, Copy plain or Copy JSON to move the result elsewhere, or Use as A to carry it into the next calculation.
Entering numbers
Cells accept whole numbers, decimals, fractions such as -1/2 and expressions like sqrt(2). Blank cells count as 0. Swap A and B switches the two matrices.
Exact fractions or decimals
With exact fractions on, every entry is kept as a fraction, so the inverse of a matrix with whole numbers comes out as clean values like −1/5 instead of −0.19999. Turn it off to work in decimals. If a cell holds something irrational, such as sqrt(2) or pi, the calculator switches to decimals for that calculation and says so.
Determinant, inverse and RREF
The determinant is found by row reducing to an upper triangular matrix and multiplying the diagonal, keeping track of row swaps, which flip the sign. The inverse comes from reducing [A | I] until the left side is the identity. RREF, reduced row echelon form, shows every swap, scaling and elimination, which is also how you read off the rank and solve systems of equations.
Eigenvalues and LU
Eigenvalues are the roots of the characteristic polynomial det(λI − A). In exact mode you get the polynomial and exact roots whenever they exist, like (5 ± √33)/2. Otherwise they are computed with the QR algorithm to about ten significant figures. Real matrices can have complex eigenvalues, which come in conjugate pairs a ± bi. LU splits a square matrix into a lower and an upper triangular matrix with a row permutation, PA = LU.
Questions
Why does it say a matrix has no inverse?
A matrix with a determinant of 0 is singular and cannot be inverted. The steps show where the rows become dependent.
Can I solve a system of equations?
Put the coefficients and the constants in one matrix and use RREF. The reduced rows give the solution. The equation solver is another option.
Why can I not multiply my matrices?
The number of columns in the first matrix must equal the number of rows in the second.
Is my data sent anywhere?
No. The calculations run in your browser.