A Matrix Determinant Calculator

A determinant collapses a square matrix into a single number that tells you how much the matrix stretches space and whether it collapses it flat.

Matrix sizePick the size of the matrix. Sarrus rule applies to the 3 by 3 case.2x2 row 1 column 1The top left entry of the 2 by 2 matrix.2x2 row 1 column 2The top right entry of the 2 by 2 matrix.2x2 row 2 column 1The bottom left entry of the 2 by 2 matrix.2x2 row 2 column 2The bottom right entry of the 2 by 2 matrix.3x3 row 1 column 1The first entry of the first row.3x3 row 1 column 2The second entry of the first row.3x3 row 1 column 3The third entry of the first row.3x3 row 2 column 1The first entry of the second row.3x3 row 2 column 2The middle entry of the matrix.3x3 row 2 column 3The third entry of the second row.3x3 row 3 column 1The first entry of the third row.3x3 row 3 column 2The second entry of the third row.3x3 row 3 column 3The bottom right entry of the matrix.

Determinant

-14

Absolute value: 14
Down-right diagonal terms: 18
Down-left diagonal terms: 32
Sum of the main diagonal: 9

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

The formula this calculator uses

2x2: det = ad - bc. 3x3: sum the three down-right diagonals and subtract the three down-left ones.

det: The determinant, a single signed number
a, b, c, d: The entries of a 2 by 2 matrix, read row by row
|det|: The area or volume scale factor of the transformation
0: A determinant of zero means the matrix is singular

How to check the result by hand

1
Confirm the matrix is square
Determinants exist only for square matrices. A 2 by 3 or 3 by 2 matrix has no determinant, so check the shape before doing any arithmetic.
2
For a 2 by 2, cross multiply and subtract
Multiply the main diagonal, 3 times 6, which is 18. Multiply the other diagonal, 8 times 4, which is 32. Subtract to get minus 14.
3
For a 3 by 3, apply Sarrus rule
Copy the first two columns to the right. The down-right products are minus 84, 10 and 32, adding to minus 42. The down-left products are minus 4, 28 and 240, adding to 264.
4
Subtract and interpret the sign
Minus 42 minus 264 is minus 306. The absolute value of 306 is the volume scale factor, and the negative sign says the transformation flips orientation.

For worked examples, common mistakes and the limits of this formula, read the full How To Calculate A Matrix Determinant page.