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The determinant of a matrix is a special value that is calculated from a square matrix.

It can help you determine whether a matrix has an inverse, find the area of a triangle, and let you know if the system of equations has a unique solution. Determinants are also used in calculus and linear algebra.

We use the notation $|A|$ or $\det(A)$ to denote the determinant of matrix A.

(Video) Determinant of 3x3 and 2x2 Matrices

by The Organic Chemistry Tutor

This video by The Organic Chemistry Tutor describes what matrix determinants are and how to find them with $2x2$ and $3x3$ matrices.

Summary

2x2 Matrix

If we're given a $2x2$ matrix, $\begin{bmatrix} a&b \\ c&d \\ \end{bmatrix}$, we want to multiply the diagonals and subtract.

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

Looking at an example,

$$\begin{bmatrix} 3&5 \\ -4&7 \\ \end{bmatrix} = (3)(7) - (5)(-4) = 21 + 20 = 41$$

3x3 Matrix

If we're given a $3x3$ matrix, $\begin{bmatrix} a_1&b_1&c_1 \\ a_2&b_2&c_2 \\ a_3&b_3&c_3 \\ \end{bmatrix}$, we can still find the determinant, but it's a little more complicated. First, we will cross out the first row and first column, leaving us with a square in the bottom right-hand corner of the matrix. We will multiply this matrix by the value in the top left-hand corner of the matrix.

$a_1 \cdot \begin{bmatrix} b_2&c_2 \\ b_3&c_3 \\ \end{bmatrix}$

Next, we will cross out the first row and second column of the matrix, and we will be left with four items. We will multiply this new matrix by the second term in the first row.

$b_1 \cdot \begin{bmatrix} a_2&c_2 \\ a_3&c_3 \\ \end{bmatrix}$

We will follow this pattern again, crossing out the first row and third column, leaving us with a matrix in the bottom left-hand corner of the matrix. We will multiply this new matrix by the term in the top right-hand corner of the matrix.

$c_1 \cdot \begin{bmatrix} a_2&b_2 \\ a_3&b_3 \\ \end{bmatrix}$

Lastly, we'll combine all of the terms into one equation to find the determinant of a $2x2$ matrix:

$$a_1 \cdot \begin{bmatrix} b_2&c_2 \\ b_3&c_3 \\ \end{bmatrix} - b_1 \cdot \begin{bmatrix} a_2&c_2 \\ a_3&c_3 \\ \end{bmatrix} + c_1 \cdot \begin{bmatrix} a_2&b_2 \\ a_3&b_3 \\ \end{bmatrix}$$

You'll use the skills you learned earlier about finding the determinant of $2x2$ matrices to find the determinant of a $3x3$ matrix.

Example

Next, he goes through an example. Try it on your own before looking at his solution!

$$\begin{bmatrix} 2&4&-3 \\ 5&7&6 \\ -8&1&9 \\ \end{bmatrix}$$

The Determinant of a $2x2$ Matrix

Before exploring determinants, it’s important to understand basic matrix operations like adding and subtracting matrices. Also make sure you have reviewed zero matrices.

In this next lesson, you will learn how to find the determinant of a $2x2$ matrix. There is a simple formula that we follow for all $2x2$ matrices.

Remember: A $2x2$ matrix has two rows and two columns. The number of rows and columns affects the formula for the determinant, so the following formula only applies with $2x2$ matrices.

The determinant is a scalar value that can give us some information about a particular matrix. We use determinants a lot when dealing with linear transformations and complex numbers.

To calculate the determinant of a $2x2$ square matrix, $\begin{bmatrix} a&b \\ c&d \\ \end{bmatrix}$, we can say that:

$$|A| = ad-bc$$

Looking Further

In some of the following lessons, you will become more comfortable with the determinant. You will especially use the determinant when solving systems of linear equations, also known as Cramer's Rule.