**What is Inverse Matrix**?

Let A be a square matrix of order *n*. A **matrix** B of order *n* is said to be the inverse of A if AB=BA=I_n

**Note**:

*The inverse of an arbitrary square matrix may not exist*.

** Example of Inverse Matrix ?**

The Matrix A=\;\begin{bmatrix}1&-1\\-1&1\end{bmatrix} has no inverse.

An inverse of a matrix, if it exists, is unique. For suppose that A has two inverses, say B and C. Then

**AB = BA=I **and ** AC = CA= I**

Therefore, by the associative law for multiplication, we have

B(AC) = BI = B

and (BA)C = IC = C

and so B = C

The unique inverse of a matrix A, if it exists, is denoted by A^{-1}

A square matrix A, whose inverse exists, is called anonsingularorinvertiblematrix. Square matrices which do not have inverse are calledsingularmatrices.

**Properties of Inverse Matrix Proof**

It is easy to verify the following properties of inverses of matrices:

**For any invertible matrix A,(A^{-1})^{-1} = A****For nonsingular matrices A and B, (AB)^{-1} = B^{-1} A^{-1}****For any invertible matrix A, (A^t)^{-1} = (A^{-1})^t**

### **Inverse Matrix** *Proof*

*Proof*

For nonsingular matrices A and B, (AB)^{-1} = B^{-1} A^{-1}.

Since A and B are nonsingular, A^{-1} and B^{-1} exist. Also, Since A and B square matrices AB are defined.

To prove that AB is nonsingular and B^{-1} A^{-1} is the inverse of AB, We show that

(AB)