Determinant of a 3 x 3 matrix

In matrices, determinants are the special numbers calculated from the square matrix. The determinant of a 3 x 3 matrix is calculated for a matrix having 3 rows and 3 columns. The symbol used to represent the determinant is represented by vertical lines on either side, such as | |. The most popular application is to find the area of a triangle using a determinant, where the three vertices of the triangle are considered as the coordinates in an XY plane.

Let A be the matrix, and then the determinant of a matrix A is denoted by |A|. To find the determinant of matrices, the matrix should be a square matrix, such as a determinant of 2×2 matrix, determinant of 3×3 matrix, or n x n matrix. It means the matrix should have an equal number of rows and columns. Finding determinants of a matrix is helpful in solving the inverse of a matrix, a system of linear equations, and so on. In this article, let us discuss how to solve the determinant of a 3×3 matrix with its formula and examples.

Determinant of a 3 x 3 Matrix Formula

We can find the determinant of a matrix in various ways. First, we have to break the given matrix into 2 x 2 determinants so that it will be easy to find the determinant for a 3 by 3 matrix.

We can find the determinant of a matrix in various ways. First, we have to break the given matrix into 2 x 2 determinants so that it will be easy to find the determinant for a 3 by 3 matrix.

Let’s suppose you are given a square matrix A where

Let’s calculate the determinant of matrix A, i.e., |A|.

By expanding along the first row elements, we get;

This can be written as:

Therefore,

Writing the determinant of a 3×3 matrix – Simple points

To find the determinant of a 3×3 matrix, first we need to find the minor matrices of any row or column elements. Suppose, we want to find the determinant of the matrix A by expanding the first column elements, then write the determinants of 2×2 matrices obtained by eliminating the corresponding elements’s row & column.

For example,

For element a₁₁, we can find the 2×2 scalar matrix as follows.

For the element a₂₁, the scalar matrix is obtained as:

Similarly, for the element a₃₁, the 2×2 scalar matrix is obtained as follows:

Thus, the determinant of matrix A is written as:

Solved Examples

Let’s understand the calculation of determinant of 3×3 matrices with the help of solved examples below:

Example 1: Calculate the determinant of the 3×3 matrix given below:

Solution:

Let’s compare the given matrix with the general matrix form of the 3×3 matrix.

That means,

Here,

a₁₁ = 1, a₁₂ = -1, a₁₃ = 0

a₂₁ = 2, a₂₂ = 3, a₂₃ = 4

a₃₁ = 0, a₃₂ = 1, a₃₃ = 2

Thus, by applying the determinant of a 3×3 matrix formula, we have

= 1 [3(2) – 1(4)] + 1[2(2) – 4(0)] + 0

= (6 – 4) + (4 – 0)

= 2 + 4

= 6

Example 2:

Find the determinant of the following 3×3 matrix.

Solution:

Let’s compare the given matrix with the general matrix form of the 3×3 matrix.

That means,

Here,

a₁₁ = 0, a₁₂ = 1, a₁₃ = 1

a₂₁ = 1, a₂₂ = 1, a₂₃ = 0

a₃₁ = 1, a₃₂ = 0, a₃₃ = 1

Now, by applying the determinant of a 3×3 matrix formula, we get;

= 0 -1 [1(1) – 0(1)] + 1[1(0) – 1(1)]

= -1(1) + (-1)

= -1 – 1

= -2

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