How to Find the Determinant of a Matrix

The determinant is a single number calculated from a square matrix. For a 2×2 matrix it is $ad - bc$. For a 3×3 matrix, expand along a row using smaller 2×2 determinants. A determinant of zero means the matrix has no inverse.

2×2 determinants

Example: $\begin{vmatrix} 4 & 7 \\ 2 & 6 \end{vmatrix} = 4 \times 6 - 7 \times 2 = 10$.

3×3 determinants: cofactor expansion

Expand along the first row with signs $+, -, +$:

Worked example

What the determinant means

• Zero determinant: the matrix is singular, it has no inverse, and its rows are linearly dependent. For example $\begin{vmatrix} 1 & 2 \\ 2 & 4 \end{vmatrix} = 0$ because the second row is double the first.
• Area scale: a 2×2 matrix maps the unit square to a parallelogram whose area is $|\det A|$.
• Sign: a negative determinant means the transformation flips orientation.

Larger matrices

For 4×4 and above, row-reduce to triangular form. The determinant is the product of the diagonal, with a sign change for each row swap.

Common mistakes

• Forgetting the alternating signs in cofactor expansion.
• Computing $ad + bc$ instead of $ad - bc$.
• Losing a sign when swapping rows.

Practice questions

1. $\begin{vmatrix} 3 & 5 \\ 1 & 2 \end{vmatrix}$
2. $\begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$
3. For which $k$ is $\begin{vmatrix} k & 2 \\ 3 & 6 \end{vmatrix} = 0$?

Answers: 1) 1 2) 6 3) $k = 1$

The determinant calculator shows the cofactor expansion and, for 2×2 matrices, animates the area change. Use it alongside how to find the inverse of a matrix and the matrix calculator.

Frequently asked questions

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