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# ISI MStat PSB 2006 Problem 1 | Inverse of a matrix This is a very beautiful sample problem from ISI MStat PSB 2006 Problem 1 based on Inverse of a matrix. Let's give it a try !!

## Problem- ISI MStat PSB 2006 Problem 1

Let A and B be two invertible real matrices. Assume that is invertible. Show that is also invertible.

### Prerequisites

Matrix Multiplication

Inverse of a matrix

## Solution :

We are given that A,B,A+B are all invertible real matrices . And in this type of problems every information given is a hint to solve the problem let's give a try to use them to show that is also invertible.

Observe that , taking determinant is both sides as A+B , A and B are invertible so |A+B| , |A| and |B| are non-zero . Hence is also non-singular .

Again we have , , taking inverse on both sides .

Now as A+B , A and B are invertible so , we have . Hence we are done .

## Food For Thought

If are non-singular matrices of the same order such that and are also non-singular, then find the value of .

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This is a very beautiful sample problem from ISI MStat PSB 2006 Problem 1 based on Inverse of a matrix. Let's give it a try !!

## Problem- ISI MStat PSB 2006 Problem 1

Let A and B be two invertible real matrices. Assume that is invertible. Show that is also invertible.

### Prerequisites

Matrix Multiplication

Inverse of a matrix

## Solution :

We are given that A,B,A+B are all invertible real matrices . And in this type of problems every information given is a hint to solve the problem let's give a try to use them to show that is also invertible.

Observe that , taking determinant is both sides as A+B , A and B are invertible so |A+B| , |A| and |B| are non-zero . Hence is also non-singular .

Again we have , , taking inverse on both sides .

Now as A+B , A and B are invertible so , we have . Hence we are done .

## Food For Thought

If are non-singular matrices of the same order such that and are also non-singular, then find the value of .

## Subscribe to Cheenta at Youtube

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