How do you find the Determinant of a 3 × 3 Matrix?

Let us understand the calculation of a 3 × 3 matrix with an example. For the given 3 × 3 matrix below.

[Tex]\begin{bmatrix} 2 & 1 & 3\\ 4 & 0 & 1\\ 2 & -1 & 2 \end{bmatrix} [/Tex]

Step 1: Choose a Reference Row or Column

Select a row and column to start, suppose in this example we take first element (2) as the reference to calculate the determinant of 3 × 3 matrix.

So, expanding along row R₁

Step 2: Cross Out Row and Column

Remove the chosen row and column in order to simplify it in a 2 × 2 matrix.

Step 3: Find the Determinant of the 2 × 2 Matrix

Find the determinant of the 2 × 2 matrix using the formula

Determinant = (a × d) – (b × c)

Here, a = 0, b = 1, c = -1, d = 2

putting these values in the above formula of determinant, we get

Determinant = (0 × 2) – (1 × -1)

Determinant = 0- (-1)

Determinant = 0+1

∴ Determinant of the 2 × 2 matrix = 1

Step 4: Multiply by the Chosen Element

Multiply the determinant of the 2 × 2 matrix by the chosen element from the reference row (which is 2,1 and 3 in this case):

first element = 2 × 1 = 2

Step 5: Repeat this process for the second element in the chosen reference row

Find the Determinant for the second element 1 by putting the values of 2×2 matrix in formula

Determinant = (a × d) – (b × c)

Here, a = 4, b= 1, c= 2, d= 2

Determinant = (4 × 2) – (1 × 2)

Determinant = 8 – 2

Determinant = 6

Now, multiply the determinant of the 2 × 2 matrix by the chosen element from the reference row (which is 1 in this case):

second element = 1 × 6 = 6

Step 6: Repeat this process for the third element in the chosen reference row

Find the Determinant for the third element 3 by putting the values of 2×2 matrix in formula

Determinant = (a × d) – (b × c)

Here, a = 4, b= 0, c= 2, d= -1

Determinant = (4 × -1) – (0 × 2)

Determinant = -4 – 0

Determinant = -4

Now, multiply the determinant of the 2×2 matrix by the chosen element from the reference row (which is 3 in this case):

second element = 3 × (-4) = -12

Step 7: Using Formula

Add up all the results from the step 4, 5, and 6

2 – 6 + (-12) = (-16)

∴ -16 is the determinant of the 3 × 3 matrix.

Determinant is a fundamental concept in linear algebra used to find a single scalar value for the given matrix. This article will explain what is a 3 × 3 Matrix and how to calculate the Determinant of a 3 × 3 Matrix step by step, as well as, its applications. Whether you are a student learning linear algebra or an enthusiast seeking a deeper understanding of matrix operations, understanding the determinant of a 3 × 3 matrix is a valuable skill to acquire.