# Properties of Inverse Matrix Proof

## 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 a nonsingular or invertible matrix. Square matrices which do not have inverse are called singular matrices.

## 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 MatrixProof

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)