Matrix Determinant Calculator — 2x2 & 3x3
Enter a 2x2 or 3x3 matrix to calculate its determinant instantly, with the exact cofactor expansion shown step by step.
Determinant
14
Expansion
det = (4 × 8) − (6 × 3) = 32 − 18 = 14
2×2 uses the direct formula ad − bc. 3×3 uses cofactor expansion along the first row: each entry in the top row is multiplied by the determinant of the 2×2 matrix left after removing its own row and column, with alternating signs (+, −, +).
About this tool
How to use it
- Pick a matrix size, 2×2 or 3×3.
- Enter each cell's value into the grid, arranged exactly like the matrix itself.
- Press Calculate Determinant, or just keep typing — the result and its expansion update live.
- Read the expansion line to see exactly how the determinant was computed, not just the final number.
Why it matters
A matrix's determinant tells you things a quick glance can't — whether the matrix can be inverted (a zero determinant means it can't), how a linear transformation scales area or volume, and whether a system of linear equations has a unique solution. Computing it by hand for a 3×3 matrix means tracking six products and their signs correctly, which is easy to get wrong.
When to use it
Useful for checking linear algebra homework, for confirming whether a matrix is singular (determinant zero, meaning it has no inverse) before relying on it in a calculation, or for verifying a determinant you've already computed by hand.
Good to know
2×2 uses the direct formula ad − bc. 3×3 uses cofactor expansion along the first row: each top-row entry is multiplied by the determinant of the 2×2 'minor' matrix left after removing that entry's own row and column, with alternating signs (+, −, +) across the row. Both methods are shown step by step in the expansion line, not just the final number, so you can check exactly where a hand calculation might have gone wrong.
Common questions
What does a determinant of zero mean?
A zero determinant means the matrix is 'singular' — it has no inverse, and if it represents a system of linear equations, that system either has no solution or infinitely many, rather than exactly one.
Why does 3×3 use a different method than 2×2?
The direct ad − bc formula only works for 2×2 matrices. Larger matrices use cofactor expansion instead, which breaks the determinant down into smaller 2×2 determinants (called minors) that can each be solved with the simple formula, then combines them with alternating signs.
Why does the expansion alternate + and − signs?
Cofactor expansion assigns each position in the matrix a sign based on a checkerboard pattern (+ − + − ...), starting with + in the top-left corner. Expanding along the first row of a 3×3 matrix uses the signs +, −, + for its three entries.
Does the order of rows and columns matter?
Yes — a determinant is defined for a specific matrix as written, and swapping two rows (or two columns) flips the sign of the determinant. Make sure your matrix is entered in the same row/column order as the one you're checking against.