# Inverse Of A Matrix

If A is a square matrix of order n and if there exists another matrix B of the same order such that
AB = BA = I, then B is called the inverse matrix of A and it is represented by A

Hence, B is the inverse of A. Or B = A

^{-1}, and A is called invertible.**Example:**Hence, B is the inverse of A. Or B = A

^{-1}and A = B^{-1}.**PROPERTIES OF INVERSE OF A MATRIX:**

**1. If the inverse of a square matrix exists, then it is unique.**

**Proof:**

Let A = [aij] be a square matrix of order n and A-1 is its inverse. And let B be another matrix which is also an inverse of A. Then,

A A-1 = A

^{-1}A = I and AB = BA= I

Multiplying both sides of AB = I by A

^{-1}, we get

A

^{-1}(AB) = A

^{-1}I = A

^{-1}

(A

^{-1}A)B = A

^{-1}

Hence IB = A

^{-1}

Hence B = A

^{-1}

Hence the inverse of a square matrix is unique.

**2. If A and B are two non-singular matrices (it means |A| ≠ 0 and |B| ≠ 0) of the same order, then AB is also not singular and (AB)**

^{-1}= B^{-1}A^{-1}, that is the inverse of the product of two non-singular matrices A and B is equal to the product of the inverses A^{-1}and B^{-1}in the reverse order.**Proof:**

According to the definition of inverse of a matrix,

(AB)(AB)

^{-1}= I

Multiplying both sides by A

^{-1}, we get

A

^{-1}(AB)(AB)

^{-1}= A

^{-1}.I

Or (A

^{-1}A).B(AB)

^{-1}= A

^{-1}(Since A

^{-1}.I = A

^{-1})

Or I.B(AB)

^{-1}= A

^{-1}

Or B(AB)

^{-1}= A

^{-1}

Or B

^{-1}B(AB)

^{-1}= B

^{-1}A

^{-1}(By multiplying both sides by B

^{-1})

Or I.(AB)

^{-1}= B

^{-1}A

^{-1}

Hence

**(AB)**(proved)

^{-1}= B^{-1}A^{-1}**Online Inverse of a Matrix Help:**

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