Elements Used to Find Inverse of 3 × 3 Matrix
There are mainly two elements used to find the Inverse of a 3 × 3 Matrix:
- Adjoint of Matrix
- Determinant of Matrix
Adjoint of a 3 × 3 Matrix
The adjoint of a matrix A is found by taking the transpose of the cofactor matrix of A. To calculate the adjoint of a matrix in detail, follow the instructions provided.
For a 3 × 3 matrix, the cofactor of any element is the determinant of a 2 × 2 matrix formed by removing the row and column containing that element. When finding cofactors, you alternate between positive and negative signs.
For example, given matrix A:
[Tex]A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & 2 & 4 \\ 1 & 1 & 2 \\ \end{bmatrix} [/Tex]
The Minor matrix is obtained as follows:
[Tex]\begin{bmatrix} \begin{vmatrix} 2 & 4 \\ 1 & 2 \\ \end{vmatrix} & \begin{vmatrix} 0 & 4 \\ 1 & 2 \\ \end{vmatrix} & \begin{vmatrix} 0 & 2 \\ 1 & 1 \\ \end{vmatrix} \\ \\ \begin{vmatrix} 1 & 3 \\ 1 & 2 \\ \end{vmatrix} & \begin{vmatrix} 2 & 3 \\ 1 & 2 \\ \end{vmatrix} & \begin{vmatrix} 2 & 1 \\ 1 & 1 \\ \end{vmatrix} \\ \\ \begin{vmatrix} 1 & 3 \\ 2 & 4 \\ \end{vmatrix} & \begin{vmatrix} 2 & 3 \\ 0 & 4 \\ \end{vmatrix} & \begin{vmatrix} 2 & 1 \\ 0 & 2 \\ \end{vmatrix} \end{bmatrix} [/Tex]
Calculate the determinants of the 2 × 2 matrices formed by multiplying diagonally and subtracting the products from left to right i.e., Minor.
[Tex]\begin{vmatrix} 2 & 4 \\ 1 & 2 \\ \end{vmatrix} [/Tex]= (2×2) – (4×1) = 4 – 4 = 0
[Tex]\begin{vmatrix} 0 & 4 \\ 1 & 2 \\ \end{vmatrix} [/Tex]= (0×2) – (4×1) = 0 – 4 = -4
[Tex]\begin{vmatrix} 0 & 2 \\ 1 & 1 \\ \end{vmatrix} [/Tex]= (0×1) – (2×1) = 0 – 2 = -2
[Tex]\begin{vmatrix} 1 & 3 \\ 1 & 2 \\ \end{vmatrix} [/Tex]= (1×2) – (3×1) = 2 – 3 = -1
[Tex]\begin{vmatrix} 2 & 3 \\ 1 & 2 \\ \end{vmatrix} [/Tex]=(2×2) – (3×1) = 4 – 3 = 1
[Tex]\begin{vmatrix} 2 & 1 \\ 1 & 1 \\ \end{vmatrix} [/Tex]=(2×2) – (1×1) = 4 – 1 = 3
[Tex]\begin{vmatrix} 1 & 3 \\ 2 & 4 \\ \end{vmatrix} [/Tex]=(1×4) – (3×2) = 4 – 6 = -2
[Tex]\begin{vmatrix} 2 & 3 \\ 0 & 4 \\ \end{vmatrix} [/Tex]=(2×4) – (3×0) = 8 – 0 = 8
[Tex]\begin{vmatrix} 2 & 1 \\ 0 & 2 \\ \end{vmatrix} [/Tex]=(2×2) – (1×0) = 4 – 0 = 4
So, the cofactor matrix is:
[Tex]\begin{bmatrix} +(0) & -(-4) & +(-2) \\ -(-1) & +(1) & -(1) \\ +(-2) & -(8) & +(4) \\ \end{bmatrix} = \begin{bmatrix} 0 & 4 & -2 \\ 1 & 1 & -1 \\ -2 & -8 & 4 \\ \end{bmatrix} [/Tex]
[Tex]\begin{bmatrix} 0 & 4 & -2 \\ 1 & 1 & -1 \\ -2 & -8 & 4 \\ \end{bmatrix} [/Tex]
By transposing the cofactor matrix, we obtain the adjoint matrix.
[Tex]\begin{bmatrix} 0 & 1 & -2 \\ 4 & 1 & -8 \\ -2 & -1 & 4 \\ \end{bmatrix} [/Tex]
Determinant of a 3 × 3 Matrix
Using the Same example as we have discussed above, we can calculate the Determinant of Matrix A
[Tex]A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & 2 & 4 \\ 1 & 1 & 2 \\ \end{bmatrix} [/Tex]
Calculate the Determinant of Matrix using the first row,
Det A = 2(cofactor of 2) + 1(cofactor of 1) + 3(cofactor of 3)
Det A = 2(0) + 1(4) + 3(-2)
Det A = 2 + 4 – 6
Det A = 0
You can check Trick to calculate determinant of a 3×3 matrix
Inverse of 3×3 Matrix
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.
Table of Content
- What is the Inverse of 3 × 3 Matrix?
- How to Find the Inverse of 3 × 3 Matrix?
- Elements Used to Find Inverse of 3 × 3 Matrix
- Inverse of 3 × 3 Matrix Formula
- Finding Inverse of 3 × 3 Matrix Using Row Operations
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