Determinant Calculator (2×2 and 3×3)

A determinant calculator finds the determinant of a square matrix, a single number that tells you whether the matrix can be inverted and how it scales area or volume. This one handles 2×2 and 3×3 matrices and shows the working for each step.

Type each row as numbers separated by commas. Leave the third row empty to calculate a 2×2 determinant, or fill all three rows for a 3×3 determinant.

How the determinant calculator works

The 2×2 determinant

For a matrix with rows (a, b) and (c, d), multiply the main diagonal and subtract the product of the other diagonal.

det = ad − bc

The 3×3 determinant

For a 3×3 matrix with rows (a, b, c), (d, e, f) and (g, h, i), expand along the first row, using the 2×2 determinant left when you cross out each entry’s row and column.

det = a(ei − fh) − b(di − fg) + c(dh − eg)

The signs alternate plus, minus, plus. Each bracket is called a minor.

The rule of Sarrus

For a 3×3 matrix you can also add the three down-right diagonals and subtract the three up-right diagonals: (aei + bfg + cdh) − (ceg + bdi + afh). It gives the same answer as the row expansion.

Worked examples

Example 1: a 2×2 matrix

For rows (4, 3) and (6, 3), det = 4 × 3 − 3 × 6 = 12 − 18 = −6. The determinant is not zero, so the matrix is invertible.

Example 2: a 3×3 matrix

For rows (2, −1, 3), (0, 4, 5) and (1, 2, −2), the three minors are 4 × (−2) − 5 × 2 = −18, 0 × (−2) − 5 × 1 = −5 and 0 × 2 − 4 × 1 = −4.

det = 2(−18) − (−1)(−5) + 3(−4) = −36 − 5 − 12 = −53

Example 3: a singular matrix

For rows (1, 2) and (2, 4), det = 1 × 4 − 2 × 2 = 0. The second row is twice the first, so the matrix has no inverse.

Useful properties

• A determinant of zero means the matrix is singular and has no inverse.
• Swapping two rows changes the sign of the determinant.
• If one row is a multiple of another, the determinant is zero.
• For a triangular matrix, the determinant is the product of the diagonal entries.

Frequently asked questions