How to Find the Determinant of a 3×3 Matrix (Step by Step)

A 3×3 determinant is a single number calculated from the nine entries of a square matrix. It tells you whether the matrix has an inverse and is used to solve systems of equations. Below are two methods — cofactor expansion, which works for any size, and the rule of Sarrus, a quick 3×3-only check — worked on the same matrix so you can see they agree.

We'll use

$$A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 4 \\ 1 & 2 & 5 \end{bmatrix}$$

First, the 2×2 determinant

Everything builds on this:

$$\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$$

For example, $\det\begin{bmatrix} -1 & 4 \\ 2 & 5 \end{bmatrix} = (-1)(5) - (4)(2) = -5 - 8 = -13$.

Method 1 — Cofactor expansion in 3 steps

Step 1 — Pick a row and use the sign pattern

Each position has a sign:

$$\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$$

We'll expand along the first row, so the signs are $+,\,-,\,+$.

Step 2 — Find each 2×2 minor

For each entry in the row, cover its row and column; the 2×2 that remains is its minor.

Step 3 — Multiply and add

$$\det A = +2(-13) - 1(-4) + 3(1) = -26 + 4 + 3 = -19$$

Method 2 — The rule of Sarrus (3×3 only)

1. Copy the first two columns to the right of the matrix.
2. Add the three downward-diagonal products: $2(-1)(5) + 1(4)(1) + 3(0)(2) = -10 + 4 + 0 = -6$.
3. Subtract the three upward-diagonal products: $3(-1)(1) + 2(4)(2) + 1(0)(5) = -3 + 16 + 0 = 13$.
4. $\det A = -6 - 13 = -19$ ✓ — the same answer.

Warning: Sarrus' rule does not extend to 4×4 or larger matrices. Use cofactor expansion (or row reduction) for those.

Shortcut: expand along the row or column with the most zeros

You can expand along any row or column — the answer is the same. Zeros save work, because their terms vanish. Column 1 of $A$ is $(2, 0, 1)$, with signs $+,\,-,\,+$:

$$\det A = 2\det\begin{bmatrix} -1 & 4 \\ 2 & 5 \end{bmatrix} - 0 + 1\det\begin{bmatrix} 1 & 3 \\ -1 & 4 \end{bmatrix} = 2(-13) + (4 + 3) = -26 + 7 = -19$$

Three methods, one answer — a strong check.

What does the determinant tell you?

• $\det A \neq 0$: the matrix is invertible, and a system of equations with this coefficient matrix has exactly one solution.
• $\det A = 0$: the matrix is singular — no inverse. For example, $\det\begin{bmatrix} 1&2&3\\4&5&6\\7&8&9 \end{bmatrix} = 0$, because the rows are linearly dependent (row 3 = 2 × row 2 − row 1).
• Geometric meaning: $|\det A|$ is the factor by which the matrix scales volume; a negative sign means orientation is flipped.
• Determinants also power Cramer's rule for solving systems — see solving simultaneous equations.

Common mistakes

• Forgetting the minus sign on the middle term of row 1.
• Mixing up $ad - bc$ (writing $bc - ad$).
• Sign slips with negative entries: $-(-1)(1) = +1$.
• Using Sarrus' rule on a 4×4 matrix.

Use the 3×3 determinant calculator

Enter your matrix in the 3×3 determinant calculator with steps to see every minor and cofactor. For smaller matrices use the 2×2 determinant calculator; to invert a 2×2, try the matrix inverse calculator. More tools are in linear algebra.

FAQ

How do you find the determinant of a 3×3 matrix? Expand along a row: multiply each entry by its signed 2×2 minor ($+,-,+$) and add the results.

What is the rule of Sarrus? A 3×3-only shortcut: copy the first two columns, add the three downward diagonal products and subtract the three upward ones.

Which row should I expand along? Any row or column gives the same result; choose the one with the most zeros.

What does a determinant of 0 mean? The matrix has no inverse, and the corresponding system of equations has either no solution or infinitely many.

Can a determinant be negative? Yes. The sign indicates orientation; the absolute value is the volume scale factor.

Is there a 3×3 determinant calculator with steps? Yes — the 3×3 determinant calculator shows the expansion.

Further reading: Wolfram MathWorld — Determinant · OpenStax College Algebra 2e.