Inverse of 3x3 Matrix

Inverse of a 3 × 3 matrix is a matrix which when multiplied by the original Matrix gives the identity matrix as the product. The 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.

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 the 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:

A =\begin{bmatrix}2 & 1 & 3 \\0 & 2 & 4 \\1 & 1 & 2 \\\end{bmatrix}

Solution :

The Minor matrix is obtained as follows:

\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}

Calculate the determinants of the 2 × 2 matrices formed by multiplying diagonally and subtracting the products from left to right i.e., Minor.

\begin{vmatrix}2 & 4 \\1 & 2 \\\end{vmatrix}= (2×2) - (4×1) = 4 - 4 = 0

\begin{vmatrix}0 & 4 \\1 & 2 \\\end{vmatrix}= (0×2) - (4×1) = 0 - 4 = -4

\begin{vmatrix}0 & 2 \\1 & 1 \\\end{vmatrix}= (0×1) - (2×1) = 0 - 2 = -2

\begin{vmatrix}1 & 3 \\1 & 2 \\\end{vmatrix}= (1×2) - (3×1) = 2 - 3 = -1

\begin{vmatrix}2 & 3 \\1 & 2 \\\end{vmatrix}=(2×2) - (3×1) = 4 - 3 = 1

\begin{vmatrix}2 & 1 \\1 & 1 \\\end{vmatrix}=(2×2) - (1×1) = 4 - 1 = 3

\begin{vmatrix}1 & 3 \\2 & 4 \\\end{vmatrix}=(1×4) - (3×2) = 4 - 6 = -2

\begin{vmatrix}2 & 3 \\0 & 4 \\\end{vmatrix}=(2×4) - (3×0) = 8 - 0 = 8

\begin{vmatrix}2 & 1 \\0 & 2 \\\end{vmatrix}=(2×2) - (1×0) = 4 - 0 = 4

So, the cofactor matrix is:

\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}

\begin{bmatrix}0 & 4 & -2 \\1 & 1 & -1 \\-2 & -8 & 4 \\\end{bmatrix}

By transposing the cofactor matrix, we obtain the adjoint matrix.

\begin{bmatrix}0 & 1 & -2 \\4 & 1 & -8 \\-2 & -1 & 4 \\\end{bmatrix}

Determinant of a 3 × 3 Matrix

Using the Same example as we have discussed above, we can calculate the Determinant of Matrix A

A =\begin{bmatrix}2 & 1 & 3 \\0 & 2 & 4 \\1 & 1 & 2 \\\end{bmatrix}

Calculate the Determinant of the 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 = 0 + 4 - 6
Det A = -2

Check: Trick to calculate the determinant of a 3×3 matrix.

The inverse of 3 × 3 Matrix Formula

To find the Inverse of a 3 × 3 Matrix A,

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 :

A =\begin{bmatrix}2 & 1 & 3 \\0 & 2 & 4 \\1 & 1 & 2 \\\end{bmatrix}

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

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

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

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 = -2.

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

A^{-1} = \begin{bmatrix} 0 & -\frac{1}{2} & 1 \\-2 & -\frac{1}{2} & 4 \\1 & \frac{1}{2} & -2 \\\end{bmatrix}

In simplifying the fractions,

A^{-1} = \begin{bmatrix}{0} & -0.5 & 1 \\ -2 & -0.5 & 4 \\ 1 & 0.5 & -2\\\end{bmatrix}

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

Also, Check

• Types of Matrices
• Invertible Matrix
• Trace of a Matrix

Solved Examples on Inverse of 3 × 3 Matrix

Example 1: Find the Inverse of

D = \begin{bmatrix}3 & 0 & 2 \\2 & 1 & 0 \\1 & 4 & 2 \\\end {bmatrix}

Solution:

Minor Matrix of D = \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}

Minor Matrix of D = \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}

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

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

Now, we will find the Determinant "D" of the original 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 = \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}

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

Example 2: Find the Inverse of

E = \begin{bmatrix}1 & 1 & 1 \\2 & 3 & 2 \\1 & 2 & 1 \\\end{bmatrix}

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

Question 1: Calculate the Inverse of the following 3×3 Matrix:

A = \begin{bmatrix}1 & 0 & 2 \\2 & 1 & 3 \\1 & 0 & 1 \\\end{bmatrix}

Question 2: Find the Inverse of Matrix B:

B = \begin{bmatrix}3 & 1 & 1 \\2 & 0 & 1 \\1 & 2 & 2 \\\end{bmatrix}

Question 3: Determine if the Matrix C is invertible and, if so, find its Inverse:

C = \begin{bmatrix}2 & 3 & 1 \\0 & 2 & 4 \\1 & 1 & 2 \\\end{bmatrix}

Question 4: Compute the Inverse of the Matrix D:

D = \begin{bmatrix}1 & 2 & 0 \\3 & 1 & 2 \\0 & 2 & 1 \\\end{bmatrix}

Question 5: For matrix E, check if it is invertible and, if it is, find its Inverse:

E = \begin{bmatrix}2 & 1 & 2 \\0 & 3 & 1 \\1 & 2 & 0 \\\end{bmatrix}