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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 A and B be n \times n real matrices such that A^{2}=A and B^{2}=B
Suppose that I-(A+B) 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 I-(A+B) is invertible which implies it's a non-singular matrix .

Now observe that ,A(I-(A+B))=A-A^2-AB= -AB as A^2=A

Again , B(I-(A+B))=B-BA-B^2=-BA as B^2=B .

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

rank(A)=rank(A(I-(A+B)))=rank(-AB)=rank(AB) and rank(B)=rank(B(I-(A+B)))=rank(-BA)=rank(BA) .

And it's also known that rank(AB)=rank(BA) . Hence rank(A)=rank(B) (Proved) .

Food For Thought

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

Similar Problems and Solutions

Related Program

ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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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 A and B be n \times n real matrices such that A^{2}=A and B^{2}=B
Suppose that I-(A+B) 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 I-(A+B) is invertible which implies it's a non-singular matrix .

Now observe that ,A(I-(A+B))=A-A^2-AB= -AB as A^2=A

Again , B(I-(A+B))=B-BA-B^2=-BA as B^2=B .

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

rank(A)=rank(A(I-(A+B)))=rank(-AB)=rank(AB) and rank(B)=rank(B(I-(A+B)))=rank(-BA)=rank(BA) .

And it's also known that rank(AB)=rank(BA) . Hence rank(A)=rank(B) (Proved) .

Food For Thought

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

Similar Problems and Solutions

Related Program

ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube

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