Maths / Linear algebra
Matrix Calculator
Add, subtract, multiply, and transpose matrices, and find the determinant, inverse, rank, and reduced row echelon form, or solve a linear system, with exact fractions and every step shown.
Matrix Calculator: All arithmetic uses exact fractions, so results such as 1/3 are never rounded. Determinants, inverses, rank, and systems of equations are found by Gaussian elimination, recording each row swap, scaling, and subtraction. A system can have one solution, none, or infinitely many, in which case each leading variable is written in terms of the free ones. Results were checked against the SymPy computer algebra system on 250 random matrices. Runs 100% locally in your browser with zero server file uploads.
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- School & study tools
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- Free · no sign-up
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- Ready to use
Runs entirely in your browser
x1 = 2
x2 = 3
x3 = -1
Show 9 steps
R1 ÷ 211/2-1/24-3-12-11-212-3R2 − -3 × R111/2-1/2401/21/21-212-3R3 − -2 × R111/2-1/2401/21/210215R2 ÷ 1/211/2-1/2401120215R1 − 1/2 × R210-1301120215R3 − 2 × R210-13011200-11R3 ÷ -110-130112001-1R1 − -1 × R310020112001-1R2 − 1 × R310020103001-1
Everything is computed with exact fractions, so results such as 1/3 are not rounded. Matrices can be up to 6 × 6.
Solving a system by hand
Write the coefficients and right-hand sides as an augmented matrix, then use three row operations, swapping two rows, multiplying a row by a non-zero number, and adding a multiple of one row to another, until each leading entry is 1 with zeros above and below. Each column without a leading 1 is a free variable.
For single fractions, the fraction calculator works step by step too; to see a two-variable system as crossing lines, plot it on the graphing calculator.
Determinants and areas
The determinant of a 2 × 2 matrix is ad − bc; its absolute value is the area of the parallelogram the two column vectors span, and in 3 × 3 it is a volume. A negative sign means the matrix flips orientation, as a mirror does.
How to use it
- Set the size of matrix A (up to 6 × 6) and type its entries; fractions like 2/3 work.
- Choose an operation; B or the vector b appears when the operation needs it.
- Read the result, and open the steps to follow each row operation.
Privacy & limitations
Everything is computed in your browser.
Related tools
Frequently asked questions
When does a matrix have no inverse?
When its determinant is 0, which happens when one row (or column) is a combination of others; the matrix then squashes space into fewer dimensions, and the operation cannot be undone.
What does the rank tell me?
The number of independent rows or columns. A system Ax = b has a solution only if adding b as a column does not raise the rank, and the solution is unique when the rank equals the number of unknowns.
Can I use decimals?
Yes: decimals are converted to exact fractions (0.25 becomes 1/4), and the Show decimals option displays results as decimals again.
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