# 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. 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**

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: