Inverse of a matrix. Here it’s all about the inverse of a matrix. Have a look!!

### The inverse of a matrix

Suppose is a square matrix. Then the inverse of the matrix is .

Also .

Here is the unit matrix or identity matrix.

Now, to get the inverse of the matrix , I will follow a few steps.

- First of all, I will find out the determinant of the matrix . That means .
- Next, I will determine the cofactor of each element of the matrix and write them in a matrix form, say, .
- Then, I will find out the transpose of the matrix . This transposed matrix will be .
- In the end, I will get the inverse of the matrix using the formula .

###### Mathematical exaplanation

Suppose is a 3 × 3 matrix like

As a first step, I’ll get the determinant of the matrix , that is, .

Next, I’ll find out the cofactor for each element of the matrix . For example, the element in the first row and the first column has the symbol .

Next element in the first row and the second column has the symbol . And it keeps going like that.

Now the cofactor of , in this case, is the determinant of the submatrix without row 1 and column 1. So that gives

In the same way, the cofactor of is the determinant of the submatrix without row 1 and column 2.

So that gives

Thus the matrix will be

Please note the sign differences in the matrix . That is very important.

Next, I’ll get the transpose of the matrix by interchanging rows and columns.In the end, I will use the formula to find out the inverse of the matrix.

Soon I will use the inverse of the matrix to solve a set of equations using matrix method.

Now I will give an example of the inverse of a matrix.

#### Example of the inverse of a matrix

Disclaimer: This is not my own example. I have chosen it from some book. I have also given the due reference at the end of the post.

Here is the example.

##### Example

According to Stroud and Booth (2013) “Find the inverse of the matrix

”

##### Solution

Here the given matrix is

First of all, I will get the determinant of the matrix.

###### Step 1

Thus will be

Now I will evaluate the determinant. This gives

Next, I will find out the matrix with the cofactors.

###### Step 2

First of all, I’ll find out the cofactors in row 1. These are and .

So I get

Now I’ll find out the cofactors in row 2. These are and .

So I get

Then

I’ll find out the cofactors in row 3. These are and .

So I get

Thus the matrix will be

As a next step, I’ll get the transpose of the matrix . And then I’ll determine the inverse of the matrix .

###### Step 3

Now to get the transpose of the matrix , I’ll interchange between its rows and columns.

Thus the transposed matrix will be

Therefore the inverse of the matrix will be

So this gives

Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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