Matrix Determinant Calculator
Compute the determinant of a 2×2 or 3×3 matrix, with the expansion shown step by step.
Step by step
- det = ad − bc
- det = (1)(5) − (2)(4)
- det = 5 − 8 = -3
How it works
The determinant is a single number that captures key facts about a matrix — whether it can be inverted, and how it scales area or volume. For a 2×2 matrix [[a, b], [c, d]], it's simply ad − bc.
For a 3×3 matrix, this tool expands along the top row: det = a(ei − fh) − b(di − fg) + c(dh − eg). Each term multiplies a top-row entry by the 2×2 determinant of the four numbers not in its row or column, with signs alternating +, −, +.
Pick 2×2 or 3×3, type in your numbers, and you'll see the result plus the intermediate products. A determinant of zero is worth noting — it means the matrix is singular and has no inverse.
Frequently asked questions
What does a determinant of zero mean?
It means the matrix is singular: it has no inverse, and its rows (or columns) are linearly dependent. Geometrically, the transformation it represents flattens space, collapsing area or volume to zero.
How is the 3×3 determinant computed?
By cofactor expansion along the first row. Each of the three top entries is multiplied by the determinant of the 2×2 block left after deleting its row and column, and the three results are combined with alternating plus and minus signs.
Does the order of rows change the determinant?
Swapping two rows flips the sign of the determinant. Other operations behave differently — for instance, adding a multiple of one row to another leaves the determinant unchanged. Order matters, so enter the matrix exactly as written.