### What Is The Need For Inverse?

Inverse means opposite of some operation performed and the result obtained is identity of that operation.

For example,

```
Additive identity
------------------
If then is additive identity.
but
(additive identity)
Therefore, subtraction is inverse operation of addition.
Multiplicative identity
-----------------------
If then is multiplicative identity because it gives as result.
but
( multiplicative identity)
Therefore, multiplying with reciprocal or division is inverse operation of multiplication.
```

The same idea can be extended to Matrix since we are unable to divide two matrices directly. If is a square matrix and invertible , then find an inverse matrix such that multiplying it with will give an identity matrix of same order.

### Related Articles

### Why Square Matrix ?

The inverse deals with negative power such as , a non-square matrix is cannot be used because it is undefined( cannot multiply).

The second reason for using square matrix is the identity matrix. An identity matrix is a square matrix only. A product of non-square matrix with its inverse will not result in an identity matrix.

If a square matrix has inverse matrix such that

Then the matrix is called **invertible matrix** and matrix is its **inverse**. If there is no for matrix , then it is called **Singular matrix.**

A matrix is singular and has no inverse if its **determinant **is 0. You will learn about determinants in future lessons.

```
Suppose is a singular matrix of order 2 x 2.
In the same manner, determinants of higher order matrices is found.
```

Therefore, only square matrix is used to find inverse which is also a square matrix of size .

### Uniqueness Of Inverse Matrix

If a square matrix is invertible, then it has exactly one inverse.

**Proof :**

Suppose that there are two inverse and for matrix . We get

```
- (1)
- (2)
```

We know that any matrix multiplied by Identity matrix will result itself. Therefore, the following is true.

```
\\by (2)
\\ by associativity property
\\ by (1)
Therefore, inverse is unique.
```

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