Inverse of 3 × 3 Matrix Formula

To find the Inverse of a 3 × 3 Matrix A, you can use the formula A-1 = (adj A) / (det A), where:

• adj A is the adjoint matrix of A.
• det A is the determinant of A.

For A-1 to exist, det A should not equal zero. This means:

• A^-1 exists when det A is not zero (A is nonsingular).
• A^-1 does not exist when det A is zero (A is singular).

Here are the steps to find the Inverse of a 3 × 3 Matrix, using the same example :

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

Step 1: Calculate the adjoint matrix (adj A).

To find the adjoint matrix, replace the elements of A with their corresponding cofactors.

[Tex]adj A= \begin{bmatrix} 0 & -1 & -2 \\ -4 & 1 & 8 \\ -2 & 1 & 4 \\ \end{bmatrix} [/Tex]

Step 2: Find the determinant of A (det A).

To calculate the determinant of A, you can use the formula for a 3 × 3 matrix. In this case, det A = -8.

Step 3: Apply the formula A^-1 = (adj A) / (det A) to find the Inverse Matrix A^-1.

Divide each element of the adjoint matrix by the determinant of A:

A^-1 = adj A/ Det A

[Tex]A^{-1} = \begin{bmatrix} -\frac{0}{8} & -\frac{-1}{8} & -\frac{-2}{8} \\ -\frac{-4}{8} & -\frac{1}{8} & -\frac{8}{8} \\ -\frac{-2}{8} & -\frac{1}{8} & -\frac{4}{8} \\ \end{bmatrix} [/Tex]

On simplifying the fractions,

[Tex]A^{-1} = \begin{bmatrix} {0} & \frac{1}{8} & \frac{1}{4} \\ \frac{1}{2} & -\frac{1}{8} & -{1} \\ \frac{1}{4} & -\frac{1}{8} & -\frac{1}{2} \\ \end{bmatrix} [/Tex]

Inverse of a 3 × 3 matrix is a matrix which when multiplied by the original Matrix gives the identity matrix as the product. Inverse of a Matrix is a fundamental aspect of linear algebra. This process plays a crucial role in solving systems of linear equations and various mathematical applications. To calculate the inverse, it is required to calculate the adjoint matrix check the matrix’s invertibility by examining its determinant (which should not equal zero), and apply a formula to derive the Inverse Matrix.

This article covers the various concepts of the Inverse of 3 × 3 Matrix and how to Find the Inverse of 3 × 3 Matrix by calculating cofactors, adjoints, and determinants of 3 × 3 Matrix. Later in this article, you will also find solved examples for better understanding, and practice questions are also provided to check what we have learned from this.