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# ISI MStat PSB 2007 Problem 2 | Rank of a matrix This is a very beautiful sample problem from ISI MStat PSB 2007 Problem 2 based on Rank of a matrix. Let's give it a try !!

## Problem- ISI MStat PSB 2007 Problem 2

Let and be real matrices such that and Suppose that is invertible. Show that rank(A)=rank(B).

### Prerequisites

Matrix Multiplication

Inverse of a matrix

Rank of a matrix

## Solution :

Here it is given that is invertible which implies it's a non-singular matrix .

Now observe that , as Again , as .

Now we know that for non-singular matrix M and another matrix N , . We will use it to get that and .

And it's also known that . Hence (Proved) .

## Food For Thought

Try to prove the same using inequalities involving rank of a matrix.

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This is a very beautiful sample problem from ISI MStat PSB 2007 Problem 2 based on Rank of a matrix. Let's give it a try !!

## Problem- ISI MStat PSB 2007 Problem 2

Let and be real matrices such that and Suppose that is invertible. Show that rank(A)=rank(B).

### Prerequisites

Matrix Multiplication

Inverse of a matrix

Rank of a matrix

## Solution :

Here it is given that is invertible which implies it's a non-singular matrix .

Now observe that , as Again , as .

Now we know that for non-singular matrix M and another matrix N , . We will use it to get that and .

And it's also known that . Hence (Proved) .

## Food For Thought

Try to prove the same using inequalities involving rank of a matrix.

## Subscribe to Cheenta at Youtube

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