Inverse of a Matrix by Elementary Operations - Matrices | Class 12 Maths

The Gaussian Elimination method is also known as the row reduction method and it is an algorithm that is used to solve a system of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients.

Inverse of a Matrix

The inverse of a matrix is a matrix A⁻¹ such that when multiplied by the original matrix A, the result is the identity matrix I. This concept is crucial for solving linear systems and matrix equations, as it allows for reversing the effects of the matrix. A matrix has an inverse only if it is square and its determinant is non-zero. Finding the inverse involves methods such as direct formulas for 2x2 matrices, or using adjoint and determinant techniques for larger matrices. The ability to compute and apply matrix inverses is fundamental in various mathematical and applied fields.

Inverse of a Matrix by Elementary Operation

The Gaussian Elimination method is also known as the row reduction method and it is an algorithm that is used to solve a system of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients. This algorithm is used to find :

• The rank of a matrix.
• The determinant of a matrix.
• The inverse of a matrix.

The operations we can perform on the matrix to modify are:

• Interchanging/swapping two rows.
• Multiplying or Dividing a row by a positive integer.
• Adding or subtracting a multiple of one row to another.

Now using these operations we can modify a matrix and find its inverse. The steps involved are:

• Step 1: Create an identity matrix of n x n.
• Step 2: Perform row or column operations on the original matrix(A) to make it equivalent to the identity matrix.
• Step 3: Perform similar operations on the identity matrix too. 

Now the resultant identity matrix after all the operations is the inverse matrix.

Examples

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

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

Examples

Note: 

Here, R1: Row 1, R2: Row 2, R3: Row 3

Example 1: Find the inverse of the following matrix by elementary operations?

Solution:

Let's perform row or column operations on the original matrix(A) to make it equivalent to the identity matrix.

Step 1: Interchange R2 and R3 rows (to make A[2][2] = 1)

Step 2: R1 = R1 + R3 (to make A[1][1] = 1)

Step 3: R2 = R2 - 3R3 (to make A[2][1] = 0)

Step 4: R3 = R3 + R1 (to make A[3][1] = 0)

Step 5: R2 = R2/-8 (to make A[2][2] = 1)

Step 6: R1 = R1 - R2 (to make A[1][3] = 0)

Step 7: R3 - 6R2 (to make A[3][2] = 0)

Step 8: R2 = R2 + R3 (to make A[2][3] = 0)

Step 9: R1 = R1 - 2R2 (to make A[1][2] = 0)

Now perform the same operation as above on the identity matrix. Result after each similar operations as above on the identity matrix, we get:

Step 1: Interchange R2 and R3 rows

Step 2: R1 = R1 + R3

Step 3: R2 = R2 - 3R3

Step 4: R3 = R3 + R1

Step 5: R2 = R2/-8

Step 6: R1 = R1 - R2

Step 7: R3 - 6R2

Step 8: R2 = R2 + R3

Step 9: R1 = R1 - 2R2

So, the inverse of matrix A is:

Example 2: Find the inverse of the following matrix by elementary operations?

Solution:

Step 1: R1 = R1 + R2

Step 2: R2 = R2 x -1

Similar operations on the identity matrix will result in:

Example 3: Find the inverse of the following matrix by elementary operations?

Solution:

Step 1: Swap R2 and R3

Step 2: R2 = R2 - R3

Similar operations on the identity matrix will result in:

Worksheet: Inverse of a Matrix

Q 1: Find the inverse of the following matrix by elementary operations? A = \begin{bmatrix} 1& 5 & 0\\ 0 & 0 & 3\\ 0 & 7 & 4 \end{bmatrix}      

Q 2:Find the inverse of the following matrix by elementary operations? A = \begin{bmatrix} 2& 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 1 \end{bmatrix}      

Q 3: Find the inverse of the following matrix by elementary operations?A = \begin{bmatrix} 1& 0 & 0\\ 0 & 0 & 3\\ 0 & 2 & 1 \end{bmatrix}      

Q 4: Find the inverse of the following matrix by elementary operations? A = \begin{bmatrix} 11& 0 & 0\\ 0 & 0 & 2\\ 0 & 2 & 0 \end{bmatrix}      

Q 5: Find the inverse of the following matrix by elementary operations? A = \begin{bmatrix} 1& 0 & 1\\ 0 & 0 & 3\\ 0 & 0 & 1 \end{bmatrix}      

Q 6: Find the inverse of the following matrix by elementary operations? A = \begin{bmatrix} 1& 1 & 0\\ 0 & 0 & 3\\ 0 & 2 & 1 \end{bmatrix}      

Summary

The inverse of a matrix is a fundamental concept in linear algebra, essential for solving systems of linear equations and various mathematical applications. For a square matrix A, its inverse A⁻¹ exists only if the matrix is non-singular, meaning its determinant is non-zero. The inverse matrix, when multiplied by the original matrix, yields the identity matrix, effectively "reversing" the effect of the original matrix. Techniques for finding the inverse include direct formulas for 2x2 matrices, the adjoint method for larger matrices, and numerical methods for practical applications. Understanding and computing matrix inverses is crucial for tasks ranging from theoretical mathematics to practical engineering and computer science.