Matrix Calculator
Calculate with exact matrices. Inspect each product cell, follow row operations and solve systems with unique, infinite or no solutions.
Calculate with exact fractions, inspect a row × column product, or follow each row operation. Enter a matrix up to 10 × 10.
A × B · 2 × 2 result
Exact result. Select a cell to see its row × column calculation.
| 58 | 64 |
| 139 | 154 |
Inside the selected cell
Row of A
Column of B
1 × 7 + 2 × 9 + 3 × 11 = 58
Row operations
Before
After
Coral marks the target row; teal marks the source row. The outlined cell is the pivot column. Augmented matrices have a divider after the columns of A.
What A does to a unit square
From entries to an exact answer
Add and subtract matrices of the same size. For A × B, each answer combines one row of A with one column of B; the inner dimensions must match. Matrix multiplication generally depends on order.
Reduced row echelon form uses row swaps, nonzero scaling and adding a multiple of another row. Rank counts pivots. For a system, the last column contains b: a zero coefficient row with a nonzero constant means no solution. Free columns produce parameters in the general solution. A zero determinant alone does not distinguish an inconsistent system from one with infinitely many solutions.
Input format, exact arithmetic and limits
Use decimal dots, integers, scientific notation or fractions (for example −2/3 entered with a normal minus sign: -2/3). Separate rows with newlines or semicolons. Separate entries with spaces or commas; do not leave empty comma cells. Each matrix may have up to 10 rows, 10 columns and 20,000 characters. Numeric coefficients may have at most 24 digits and exponents from −12 to 12.
Calculations use reduced integer fractions, with no floating-point cutoff for rank or singularity. Intermediate fractions are limited to 4096 digits and calculations to 50,000 arithmetic operations. Excessive complexity produces an error. If row working alone exceeds 250,000 characters, the complete exact answer remains available with an explicit notice. Approximate display rounds to 6 significant digits and never changes the exact answer or copied values.
Inverse calculations reduce [A | I] and verify both A A⁻¹ = I and A⁻¹ A = I. System results verify A times the particular solution equals b and A times every parameter vector equals zero. For a 2 × 2 matrix, det(A) = a₁₁a₂₂ − a₁₂a₂₁; its absolute value is the transformed unit square's area.
