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 !!
Let and
be
real matrices such that
and
Suppose that is invertible. Show that rank(A)=rank(B).
Matrix Multiplication
Inverse of a matrix
Rank of a matrix
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) .
Try to prove the same using inequalities involving 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 !!
Let and
be
real matrices such that
and
Suppose that is invertible. Show that rank(A)=rank(B).
Matrix Multiplication
Inverse of a matrix
Rank of a matrix
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) .
Try to prove the same using inequalities involving rank of a matrix.