Application of Determinant of a 3 × 3 Matrix

Determinant of a Matrix can be used to find the inverse and solve the system of linear equation. Hence, we learn to find the inverse of 3 × 3 Matrix and also solve system of linear equation using Cramer’s Rule which involve the use of determinant of 3 × 3 Matrix.

Inverse of 3 × 3 Matrix

The formula to find the inverse of a square matrix A is:

[Tex]A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) [/Tex]

Where,

• A-1 is the inverse of matrix A.
• Det(A) represents the determinant of matrix A.
• adj(A) stands for the adjugate of matrix A

In simple terms, you can follow these steps to find the inverse of a matrix:

Step 1. Calculate the determinant of matrix A.

Step 2. Find the adjugate of matrix A.

Step 3. Multiply each element in the adjugate by 1/det(A).

This formula is used for square matrices (matrices with the same number of rows and columns) and assumes that the determinant is non-zero, which is a necessary condition for a matrix to have an inverse.

Cramer’s Rule

Cramer’s Rule provides a formula to solve a system of linear equations using determinants. For a system of linear equations with n variables are given in the form of

AX=B

Where,

• A = Coefficient of the square matrix
• X = Column matrix having variables
• B = Column matrix having constants

Consider the following system of linear equation

a₁x + b₁y + c₁z + . . . = d₁

a₂x + b₂y + c₂z + . . . = d₂

. . .

a_nx + b_ny + c_nz + . . . = d_n

The variables x, y, z, …, are determined using the following formulas:

• x = D_x/D
• y = D_y/D
• z = D_z/D

Where:

• D is the determinant of the coefficient matrix.
• D_x is the determinant of the matrix obtained by replacing the coefficients of x with the constants on the right-hand side.
• D_y is the determinant of the matrix obtained by replacing the coefficients of y
• D_z is the determinant of the matrix obtained by replacing the coefficients of z

Cramer’s Rule is applicable when the determinant of the coefficient matrix D is non-zero. If D = 0, the rule cannot be applied which indicates either no solution or infinitely many solutions depending on the specific case.

Also, Check

• Types of Matrices
• System of Linear Equations with Three Variables
• Matrix Operations

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.