In the previous article, you learned about inverse of matrix and why it is important. You will learn how to find inverse of a matrix in this article. There are two primary method of finding inverse of any square invertible matrix – **Classical adjoint method and Gauss -Jordan elimination method.**

**Methods To Find Inverse Of Matrix**

The primary method of finding of inverse of matrix are

- Classical adjoint method
- Gauss-Jordan elimination method

Throughout this article, we will discuss about these two methods in detail.

### Classical Adjoint Method

If is an invertible matrix then we can find the inverse of matrix with the adjoint of matrix .

But,before we begin you must understand a few terminologies.

**Determinant** – It is a special number obtained from a square matrix, non-square matrix do not have determinants. If is a square matrix then there is a number of ways to denote its determinant.

```
or or
```

**Minor of a matrix** – The minor of a square matrix is determinant obtained by deleting a row and a column from the determinant of a larger square matrix. It is denoted by for element where is the i^{th}row and is the j^{th} column.

```
The determinant of 2 x 2 matrix
You get following matrix of determinants
```

**Cofactor of a Matrix **– The cofactor of a matrix can be found from determinant of minors by assigning appropriate negative or positive signs. Each cofactor can be found by following equation.

```
The matrix of cofactors of A
```

**Adjoint of a Matrix **– The adjoint of a matrix is obtained by transposing the co-factor matrix. It is denoted as .

### Finding Inverse Of Matrix Through Adjoint Method

The process of finding inverse of matrix using adjoint method is as follows.

- Find the matrix of minors
- Find the matrix of co-factors
- Find the determinant det(A) by multiplying first row of matrix A with first row of co-factor matrix of A.
- Find the adjoint matrix usign co-factor matrix of A.
- Multiply 1/det(A) with adjoint of A (adj A) to get the inverse matrix of A.

We will find inverse of a matrix using the adjoint of matrix in the next section. First we must find the inverse of matrix, then and finally matrix.

**Example #1** : Find the inverse of matrix using adjoint method.

**Solution: **

```
Let be a matrix of order 1 x 1.
Step 1: There is no minor for 1x1 matrix.
Step 2: There is no cofactor for 1x1 matrix.
Step 3: The determinant of A is .(invertible)
Step 4: The adjoint of matrix A is .
To compute the inverse of matrix A use
```

**Example #2 **: Find the inverse of matrix using adjoint method.

**Solution:**

Let be a square and invertible matrix of order .

**Step 1: **The minors of matrix are

```
The minor matrix is
Step 2: The cofactor of matrix is obtained from minor matrix.
The cofactor matrix is
Step 3: To compute the determinant simply multiply corresponding elements of top row of matrix and tow row of cofactor of and add them.
Top row of matrix B = 2, 1
Top row of cofactor matrix = 7 -3
Step 4: Find the adjoint of the matrix by transposing cofactor matrix.
To find the inverse of matrix use following
Verify the inverse of matrix
```

**Example #3** : Find the inverse of 3 x 3 matrix using the adjoint method.

**Solution :**

Given the matrix of order .

.

**Step 1:** Find the minors of the matrix .

```
We have the following matrix of minors
```

**Step 2: **Find the cofactors of matrix .

```
We can use the matrix of minors to find the matrix of cofactors.
We get the cofactor matrix of
```

**Step 3: **Find determinant of the matrix .

```
Top row of matrix A = 1, -2, 1
Top row of coactor of A = 5, -18, 2
```

**Step 4: **Find the adjoint of matrix . The adjoint of the matrix can be obtained from transposing the cofactor matrix.

```
```

Step 5: Find the inverse of matrix using following equation.

```
```

**Verify results**:

```
Verify that
```

### Finding Inverse of Matrix Using Gauss-Jordan Elimination Method

The Gauss-Jordan elimination method convert a matrix into reduced-row echelon form to find the value of solution vector in , but here we use the technique to find inverse matrix .

**Row Operations **

The Gauss-Jordan technique involves row operations on augmented matrix obtained from the system of linear equations that transforms the matrix into identity matrix where is the order of matrix .

These row operations are

- Multiply a row with scalar where .
- Interchange row with row .
- Addition of times row to row .

**Example #5 :** Row operations

```
Let be matrix of order .
Multiply a row 1 with 3.
Interchange row 2 with row 1.
Add 2 times row 1 to row 2.
```

**Elementary Matrix **

An elementary matrix is a matrix obtained from performing a single row operation on identity matrix .

Suppose A is a matrix of order , then the product is same as performing that row operation on matrix .

**Example #6 : Elementary Matrix**

```
Suppose is a matrix of order .
Let be row operation on of order .
Let
Let us perform on .
(1)
Now, let us find matrix.
(2)
Therefore, (1) and (2) are same.
```

**Inverse Operations**

If matrix is a result of row operation on identity matrix , then there exist some operation if performed on will give back . Such an operation is called *an Inverse operation.*

For every elementary row operation, there is an equivalent inverse operation. Check the table below.

Elementary Row Operation | Inverse Operation |

Multiply row i by c where c != 0 | Multiply row i by 1/c |

Interchange row i with row j | Interchange row j with row i |

Add k times row i to row j | Add -k times row i to row j |

**Example #7** : Inverse Operations

```
Multiply row by .
Multiply row by where
```

From the above we can conclude that

```
1. Some operation on give the elementary matrix .
2. The inverse of previous operation on E will give back .
3. Inverse operation on will result in an inverse matrix such that
Note that is invertible matrix and has determinant greater than 0.
```

**4 Necessary Statements For Gauss-Jordan technique** **and inverses**

Before we find inverse of a matrix using Gauss-Jordan technique, there are 4 necessary prepositions that must be true about the matrix and its inverse.

If matrix is square matrix these 4 statement must always be true.

- Matrix is invertible meaning the determinant is greater than 0 and does not have a zero row.
- has only trivial solution.
- The matrix is reduced to reduced-row echelon form which is identity matrix . This is because has trivial solution only, which means has one solution for every .

- The matrix is product of elementary matrices.

```
Multiplying A with elementary matrices is same as performing row operations on A which will reduce the matrix to identity matrix .
Similarly, each of the is invertible, a series of inverse operation on I will give back .
also,
```

Therefore, the matrix A$ is product of elementary matrices.

**Example #8:** Find the inverse of following matrix using Gauss-Jordan elimination method.

**Solution:**

Given the matrix we will find the inverse of matrix using the Gauss-Jordan elimination method.

```
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Step 7:
Step 8:
```

The matrix is reduced to an identity matrix and the identity matrix after Gauss-Jordan elimination method reduced to inverse matrix of .

The verification of resultant inverse matrix is left as an exercise for you.