Pick the size.
2×2, 3×3 or 4×4 — the bracketed grid redraws itself, and values you've already typed carry over to the overlapping corner. No menus, no modes, just the matrix.
3×3 → NINE CELLS
01
The instrument.
Pick a size, fill the bracketed grid — the determinant solves on every keystroke. The big line is the answer; beneath the screen, the complete written proof: the cofactor expansion, minor by minor, sign by sign — and a second, independent method that must agree before anything is called a result.
BLANKS COUNT AS 0 · DECIMALS BECOME EXACT FRACTIONS
TRY
THE WORK — STEP BY STEP
LIVE
SWS-DC/01·COFACTOR EXPANSION·ELIMINATION CHECK·SARRUS FOR 3×3·0 TRACKING
02
How to use it.
Ninety seconds to full fluency — the interface is a matrix in brackets. Fill it, and the linear algebra writes itself underneath.
Fill the grid.
Type straight into the cells. Blanks count as 0, negatives welcome, and decimals become exact fractions internally — 0.5 is stored as 1/2, so nothing drifts.
0.5 → 1⁄2 INSIDE
No equals needed.
The determinant solves live on every keystroke. The echo line reads det(A) = with your matrix rendered in brackets beside it — question and answer, together.
TYPE → SOLVED
Follow the expansion.
THE WORK writes the cofactor method the way a margin would: the row or column chosen (the one with the most zeros), each minor with its own arithmetic, the signs, the alternating sum.
M₁₁ = 3×(−2) − 0×5 = −6
Read the verdict.
Below the answer: invertible or singular, the decimal form when the result is a fraction, and the sign of the value — positive stretch, negative flip, zero collapse.
det ≠ 0 → INVERTIBLE
Trust the checks.
Every determinant is recomputed independently by Gaussian elimination — the pivot product must land on the same number — and 3×3 matrices get Sarrus' rule as a third witness.
2 METHODS · 1 ANSWER ✓
studywithstem:worked-example
ONE MATRIX
det [[2, 0, 1], [1, 3, 0], [0, 5, −2]]
├── expandrow 1 · signs + − +
├── M₁₁3×(−2) − 0×5 = −6
├── M₁₂1×(−2) − 0×0 = −2 · ×0, drops
├── M₁₃1×5 − 3×0 = 5
├── combine2×(−6) − 0×(−2) + 1×5 = −7
└── checksarrus ✓ · pivots 2 × 3 × −7⁄6 ✓
RESULT −7 · INVERTIBLE · 2 INDEPENDENT CHECKS
03
Frequently asked.
The questions that keep coming back about determinants — answered once, in full.
Q1 What is a determinant, exactly?
A determinant is a single number computed from a square matrix. Geometrically, it is the signed scale factor of the transformation the matrix represents: for a 2×2 it is the factor by which area stretches (the sign says whether space flips), for a 3×3 it is volume. Algebraically it decides whether a matrix can be inverted — and it appears throughout Cramer's rule and eigenvalue equations.
Q2 How do I calculate the determinant of a 2×2 matrix?
Multiply down the main diagonal, subtract the anti-diagonal: det = ad − bc. For [[3, 8], [4, 6]]: 3×6 − 8×4 = 18 − 32 = −14. The tool shows both diagonal products and the subtraction, every time.
Q3 How do I calculate the determinant of a 3×3 matrix?
Cofactor expansion along a row: det = a(ei − fh) − b(di − fg) + c(dh − eg), where each bracketed pair is the 2×2 minor left after crossing out that entry's row and column. Pick the row or column with the most zeros to multiply less. Sarrus' rule — copy the first two columns to the right, add the ↘ products, subtract the ↗ ones — is a handy cross-check the tool runs automatically.
Q4 How do I calculate the determinant of a 4×4 matrix?
The same way, one level up: expand along the row or column with the most zeros, which reduces the 4×4 to a signed sum of 3×3 minors, then evaluate each 3×3 by its own expansion. The tool does exactly this — it scans all eight lines, picks the smartest, and shows every minor, its value, and the signed sum.
Q5 What does it mean when the determinant is 0?
The matrix is singular: it has no inverse, its rows (and columns) are linearly dependent — one is a combination of the others — and geometrically the transformation squashes space into a lower dimension. A matrix like [[1,2,3],[2,4,6],[7,8,9]], where row 2 is twice row 1, cannot be undone. Try the SINGULAR example chip.
Q6 Can a determinant be negative?
Yes. The magnitude is the scale factor — how much area or volume grows — and the sign records orientation: a negative determinant means the transformation flips space, like a mirror. In 2D, a negative determinant swaps clockwise for counterclockwise.
Q7 What determinant rules should I know?
The essentials: swapping two rows flips the sign; scaling a row scales the determinant; adding a multiple of one row to another leaves it unchanged; two identical rows force det = 0; triangular matrices have det equal to the product of the diagonal; det(I) = 1; det(AB) = det(A)·det(B); and det(Aᵀ) = det(A).
Q8 Does it handle decimals, and is it free?
Yes to both. Decimal entries become exact fractions internally — 0.1 is stored as 1/10 — so results never drift, and fractional results display as both an exact fraction and a decimal. The tool is completely free, requires no sign-up, runs entirely in your browser, and keeps working offline once loaded.