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.

What is the Inverse of 3 × 3 Matrix?

The Inverse of a 3 × 3 Matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. To find the Inverse, you can calculate the adjoint matrix, determine if the matrix is invertible (non-singular) by checking its determinant (which should not equal zero), and then apply the formula A^-1 = (adj A) / (det A). The Inverse Matrix allows you to solve systems of linear equations and perform various mathematical operations.

How to Find the Inverse of 3 × 3 Matrix?

Follow the steps given below in order to find the Inverse of 3 × 3 Matrix:

Step 1: Firstly, verify if the matrix can be inverted. To do this, calculate the determinant of the matrix. If the determinant is not zero, then proceed to the next step.

Step 2: Calculate the determinant of smaller 2 × 2 matrices within the larger matrix.

Step 3: Create the cofactor matrix.

Step 4: Obtain the Adjugate or Adjoint of the matrix by making the transpose of the cofactor matrix.

Step 5: Finally, divide each element in the adjugate matrix by the determinant of the original 3 by 3 matrix.

Related Read

• Cofactor and Minors of Matrix
• Transpose of Matrix

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 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]

Finding Inverse of 3 × 3 Matrix Using Row Operations

To find the inverse of a 3×3 Matrix, you can follow these steps:

Step 1: Start with the given 3×3 Matrix A and create an identity matrix I of the same size, placing A on the left side and I on the right side of an augmented matrix, separated by a line.

Step 2: Apply a series of row operations to the augmented matrix on the left side to transform it into the identity matrix I. The matrix on the right side of the line, which becomes A^-1, is the Inverse of the original matrix A.

Learn More, Elementary Operation of Matrices

Solved Examples on Inverse of 3 × 3 Matrix

Example 1: Find the Inverse of

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

Solution:

Minor Matrix of D = [Tex]\begin{bmatrix}\begin{pmatrix}1&0\\4&2\end{pmatrix}&\begin{pmatrix}2&0\\1&2\end{pmatrix}&\begin{pmatrix}2&1\\1&4\end{pmatrix}\\\begin{pmatrix}0&2\\4&2\end{pmatrix}&\begin{pmatrix}3&2\\1&2\end{pmatrix}&\begin{pmatrix}3&0\\1&4\end{pmatrix}\\\begin{pmatrix}0&2\\1&0\end{pmatrix}&\begin{pmatrix}3&2\\2&0\end{pmatrix}&\begin{pmatrix}3&0\\2&1\end{pmatrix}\end{bmatrix} [/Tex]

Minor Matrix of D = [Tex]\begin{bmatrix}\left(2-0\right)&\left(4-0\right)&\left(8-1\right)\\\left(0-8\right)&\left(6-2\right)&\left(12-0\right)\\\left(0-2\right)&\left(0-4\right)&\left(3-0\right)\end{bmatrix} [/Tex]

Cofactor of Matrix i.e., X = [Tex]\begin{bmatrix}+2&-\left(-4\right)&+7\\-\left(-8\right)&+4&-\left(12\right)\\+2&-\left(-4\right)&+3\end{bmatrix} [/Tex]

Transpose of Matrix X = Adj D = [Tex]\begin{bmatrix}2&8&2\\-4&4&4\\7&-12&3\end{bmatrix} [/Tex]

Now, we will find the Determinant of D using the first row:

Det D = 3(2) + 0(-4) + 2(7)

⇒ Det D = 6+0+14

⇒ Det D = 20

Inverse of Matrix D or D^-1 = Adj D / Det D

⇒ D^-1 = [Tex]\begin{bmatrix}\frac{2}{20}&\frac{8}{20}&\frac{2}{20}\\-\frac{4}{20}&\frac{4}{20}&\frac{4}{20}\\\frac{7}{20}&-\frac{12}{20}&\frac{3}{20}\end{bmatrix} [/Tex]

⇒ D^-1 = [Tex]\begin{bmatrix}\frac{1}{20}&\frac{2}{5}&\frac{1}{10}\\-\frac{2}{5}&\frac{2}{5}&\frac{2}{5}\\\frac{7}{20}&-\frac{3}{5}&\frac{3}{20}\end{bmatrix} [/Tex]

Example 2: Find the Inverse of

[Tex]E = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \\ \end{bmatrix} [/Tex]

Minor of the Matrix E = [Tex]\begin{bmatrix}\begin{pmatrix}3&2\\2&1\end{pmatrix}&\begin{pmatrix}2&2\\1&1\end{pmatrix}&\begin{pmatrix}2&3\\1&2\end{pmatrix}\\\begin{pmatrix}1&1\\2&1\end{pmatrix}&\begin{pmatrix}1&1\\1&1\end{pmatrix}&\begin{pmatrix}1&1\\1&2\end{pmatrix}\\\begin{pmatrix}1&1\\3&2\end{pmatrix}&\begin{pmatrix}1&1\\2&2\end{pmatrix}&\begin{pmatrix}1&1\\2&3\end{pmatrix}\end{bmatrix} [/Tex]

Cofactor of Matrix E i.e., X = [Tex]\begin{bmatrix}\left(3-4\right)&\left(2-2\right)&\left(4-3\right)\\\left(1-2\right)&\left(1-1\right)&\left(2-1\right)\\\left(2-3\right)&\left(2-2\right)&\left(3-2\right)\end{bmatrix} [/Tex]

X= [Tex]\begin{bmatrix}-1&0&1\\1&0&-1\\-1&0&1\end{bmatrix} [/Tex]

Adj E = [Tex]\begin{bmatrix}-1&1&-1\\0&0&0\\1&-1&1\end{bmatrix} [/Tex]

Let us now find Determinant of Matrix E using the first row:

Det E = 1(-1) + 1(0) + 1(1)

Det E= -1 + 0 + 1

Det E = 0

∴ As the determinant of the matrix E is equivalent to 0, the Inverse of Matrix E or E^-1 is not possible.

Practice Questions on Inverse of 3 × 3 Matrix

Q1. Calculate the Inverse of the following 3×3 Matrix:

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

Q2. Find the Inverse of Matrix B:

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

Q3. Determine if the Matrix C is invertible and, if so, find its Inverse:

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

Q4. Compute the Inverse of the Matrix D:

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

Q5. For matrix E, check if it is invertible and, if it is, find its Inverse:

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

Inverse of 3×3 Matrix – FAQs

1. What is the Inverse of a 3×3 Matrix?

The Inverse of a 3×3 Matrix is another matrix that, when multiplied by the original matrix, yields the identity matrix.

2. Why is finding the Inverse important?

It is essential for solving systems of linear equations, transformations, and various mathematical operations.

3. How do you calculate the Inverse of a 3×3 Matrix?

You typically find the adjoint matrix, check the determinant’s non-zero value, and apply a specific formula.

4. When does the Inverse of a 3×3 Matrix not exist?

It doesn’t exist when the determinant of the matrix is zero, making it singular.

5. Can any 3×3 Matrix have an Inverse?

No, only non-singular matrices with a non-zero determinant have inverses.

6. What is the role of the Adjoint Matrix in finding the Inverse?

The adjoint matrix helps in calculating the inverse by providing cofactors for each element.

7. In which fields is the concept of 3×3 Matrix inversion widely used?

The concept of 3×3 Matrix inversion is used in engineering, physics, computer graphics, and various mathematical disciplines.

8. How to get Inverse of 3×3 Matrix?

To find the inverse of a 3×3 matrix, you can follow these steps:

• First, calculate the determinant of the matrix.
• If the determinant is not equal to 0, proceed to the next step. If it’s 0, the matrix doesn’t have an inverse.
• Find the matrix of minors by creating 3×3 matrices for each element in the original matrix, excluding the row and column of the element you are focusing on.
• Calculate the matrix of cofactors by applying a pattern of plus and minus signs to the elements of the matrix of minors.
• Transpose the matrix of cofactors by swapping rows with columns.
• Finally, divide the transposed matrix of cofactors by the determinant to get the inverse of the 3×3 matrix.