# The Formula of the Determinant of 3×3 Matrix

The standard formula to find the determinant of a 3×3 matrix is a break down of smaller 2×2 determinant problems which are very easy to handle. If you need a refresher, check out my other lesson on how to find the determinant of a 2×2. Suppose we are given a square matrix $A$ where,

The determinant of matrix A is calculated as

Here are the key points:

• Notice that the top row elements namely $a$, $b$ and $c$ serve as scalar multipliers to a corresponding 2-by-2 matrix.
• The scalar $a$ is being multiplied to the 2×2 matrix of left-over elements created when vertical and horizontal line segments are drawn passing through $a$.
• The same process is applied to construct the 2×2 matrices for scalar multipliers $b$ and $c$.

The practice problems below can help you see if you already know how to find the determinant of a 3×3 matrix. Feel free to use it as an additional study material.

Determinant of 3×3 Matrix Practice Problems

## Determinant of 3 x 3 Matrix (animated)

### Examples of How to Find the Determinant of a 3×3 Matrix

Example 1: Find the determinant of the 3×3 matrix below.

The set-up below will help you find the correspondence between the generic elements of the formula and the elements of the actual problem.

Applying the formula,

Example 2: Evaluate the determinant of the 3×3 matrix below.

Be very careful when substituting the values into the right places in the formula. Common errors occur when students become careless during the initial step of substitution of values.

In addition, take your time to make sure your arithmetic is also correct. Otherwise, a single error somewhere in the calculation will yield a wrong answer in the end.

Since,

our calculation of the determinant becomes…

Example 3: Solve for the determinant of the 3×3 matrix below.

The presence of zero (0) in the first row should make our computation much easier. Remember, those elements in the first row, act as scalar multipliers. Therefore, zero multiplied by anything will result in the entire expression to disappear.

Here’s the setup again to show the corresponding numerical value of each variable in the formula.

Using the formula, we have…

Example 4: Solve for the determinant of the 3×3 matrix below.

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

Solution:

Example 5: Calculate the determinant of the three-by-three matrix below.

\begin{bmatrix} -5 & -5 & -5 \\ 3 & -1 & -2 \\ 4 & 2 & 1 \end{bmatrix}

Solution:

You may also be interested in these related math lessons or tutorials:

Determinant of 3×3 Matrix Problems with Answers

Determinants of a 2×2 Matrix

Determinant 2×2 Matrix Problems with Answers

Cramer’s Rule 2×2

Cramer’s Rule 3×3