What is the Matrix Determinant?

The determinant is a number we can calculate from a square matrix. It describes certain properties of the matrix and can be used for solving linear systems of equations. The determinant of a matrix is notated with vertical bars similar to absolute value notation. For example, the determinant of a matrix A is notated as |A|.

The method for finding the determinant depends on the size of the matrix. In this lesson, we will show how to find the determinant of 1x1, 2x2, and 3x3 matrices. A 4x4 or larger matrix takes very long to calculate by hand and can be very tedious. For matrices of that size, we recommend using a calculator such as the Determinant Calculator.

How to Find the Determinant of a 1x1 Matrix

The determinant of a 1x1 matrix is the number inside the matrix.

For example, if matrix A = [5] then the determinant is |A| = 5.

How to Find the Determinant of a 2x2 Matrix

$$\text{For a 2x2 matrix, if matrix A = } \begin{bmatrix} a & b \\ c & d \end{bmatrix} \text{ then the determinant is } \textbf{|A| = ad - bc}\text{.}$$

An easy way to remember the order of components that get multiplied and subtracted is to imagine the diagonal line and direction that the components are on.

The 2x2 determinant goes top left to bottom right minus top right to bottom left. That order of diagonal and direction gives us ad - bc.

How to Find the Determinant of a 3x3 Matrix

$$\begin{align} & \text{If matrix A = } \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \\ \\ & \text{then the determinant |A| = } a\begin{bmatrix} e & f \\ h & i \end{bmatrix} - b\begin{bmatrix} d & f \\ g & i \end{bmatrix} + c\begin{bmatrix} d & e \\ g & h \end{bmatrix} \text{.} \end{align}$$

The top row components of the 3x3 matrix are always the leading coefficients on each smaller 2x2 determinant within the formula. Make sure to remember that there is a negative sign in front of b!

A simple way to remember the 3x3 matrix determinant is to imagine it as three 2x2 matrix determinants. The top row components are leading coefficients in front of each 2x2 determinant.

Break the 3x3 matrix down into the top row 1x3 matrix, and the bottom two rows as a 2x3 matrix. The 2x3 matrix has three combinations of 2x2 matrices in it. Simply memorize the pattern of 1x3 components that multiply by the three 2x2 matrix determinants.