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# Inverse Matrix Calculator

To find the inverse of matrix, choose the order of the matrix, input the entries of the matrix, and click calculate button using inverse matrix calculator

## Matrix Inverse Calculator

Inverse matrix calculator solves a matrix to find its reverse using the elementary row and column elimination. It will give the whole process of finding the inverse of the matrix.

## What is the inverse of the matrix?

The reverse of the original matrix is called its inverse. It is often denoted by A-1. When a matrix is multiplied by its inverse, it gives the identity matrix. This trick can help to verify the accuracy of the inverse matrix.

## Matrix inverse formula:

The formula used for the inverse matrix is:

A^{-1 }= (1/ |A|) . adj A

Where

- |A| is the determinant of the matrix A.
- Adj A means adjoint of A. It is found using transpose.

This formula helps in a 2x2 square matrix better. For 3x3 or bigger matrices, mostly the elementary transformation is used.

## How to calculate the inverse of a matrix?

There are basically four methods, two of which are discussed above. The list of these methods is:

- Inverse matrix calculator.
- Using the formula.
- Elementary transformations.
- Finding the cofactors.

The calculator uses the second last method. There are two ways to carry out this transformation; row elimination and column elimination. The rows are subtracted or added and multiplied to each other and in some cases both.

- Write the equation
**A=I.A**, where I is the identity matrix of the same order. - The transformation will be applied on the
**A**at the LHS and**I**on the RHS. the**A**on the RHS will remain intact. - Perform different techniques on the Left-hand side
**A**to make it an identity matrix. Make the same changes to the identity matrix regardless. - When the
**A**has become the identity matrix, the original identity matrix**I**would now be different and called matrix**B**. That is**I=BA**. **I=BA**is equal to**A**, hence^{-1}=B**B**is the inverse of matrix**A**.

**Example: (row elimination)**

Find the inverse of the following matrix:

**Solution:**

**Step 1:** Write the formula.

A=l.A

**Step 2:** Subtract the R_{1} multiplied by 3 from R_{3}.

**Step 3:** Divide R_{2} by 2.

**Step 4:** Subtract R_{2} multiplied by 3 from R_{1}.

**Step 5: **Multiply R_{2} by 8 and add it to R_{3}.

**Step 6: **Divide R_{3} by 5.

**Step 7:** Multiply R_{3} by 2 and add it to R_{1}.

**Step 8: **Subtract R_{3} from R_{2}.

An identity matrix on LHS is obtained. Hence I = BA or A-1 = B.