57 352
Assignments Done
Successfully Done
In February 2018
Your physics homework can be a real challenge, and the due date can be really close — feel free to use our assistance and get the desired result.
Be sure that math assignments completed by our experts will be error-free and done according to your instructions specified in the submitted order form.
Our experts will gladly share their knowledge and help you with programming homework. Keep up with the world’s newest programming trends.

Answer on Linear Algebra Question for Matthew Lind

Question #23765
Let A and B be M by N matrices, P an invertible M by M matrix, and Q an invertible N by N matrix, such that B = PAQ, that is, the matrices A and B are equivalent. Show that the rank of B is the same as the rank of A. (Show that A and AQ have the same rank).
Expert's answer
Every invertible matrix is afinite product of some elementary matrices, that are in one to one
correspondence with elementary row and column operations of matrix. Thus, as it
is known, every elementary row/column operation preserves rank of matrix. So, matrix

Q=E1*E2*...*En, where each Ei is elementary matrix, AQ=(...((A*E1)*E2)*...)*En
and in every bracket rank is equal rank of matrix A.

So rank(A)=rank(AQ).
As rank(A^T)=rank(A), then as P^T is also invertible,
rank(PAQ)=rank(Q^T * A^T * P^T)=rank(Q^T * A^T)=rank(AQ)=rank(A) and we are

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!


No comments. Be first!

Leave a comment

Ask Your question