Question

Q.Let A and B be n × n matrices such that A = A^2, B =...


Q.Let A and B be n × n matrices such that A = A^2,
B = B^2, and AB = BA = 0.
(a) Prove that rank(A + B) = rank(A) + rank(B).
(b) Prove that rank(A) + rank(In − A) = n.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
4.4.3. Suppose A and B are n × n matrices. Prove that, if AB is invertible,...
4.4.3. Suppose A and B are n × n matrices. Prove that, if AB is invertible, then A and B are both invertible. Do not use determinants, since we have not seen them yet. Hint: Use Lemma 4.4.4. Lemma 4.4.4. If A ∈ Mm,n(F) and B ∈ Mn,k(F), then rank(AB) ≤ rank(A) and rank(AB) ≤ rank(B).
Prove the following statements: a) If A and B are two positive semidefinite matrices in IR...
Prove the following statements: a) If A and B are two positive semidefinite matrices in IR ^ n × n , then trace (AB) ≥ 0. If, in addition, trace (AB) = 0, then AB = BA =0 b) Let A and B be two (different) n × n real matrices such that R(A) = R(B), where R(·) denotes the range of a matrix. (1) Show that R(A + B) is a subspace of R(A). (2) Is it always true...
Let A, B be n × n matrices. The following are two incorrect proofs that ABhas...
Let A, B be n × n matrices. The following are two incorrect proofs that ABhas the same non-zero eigenvalues as BA. For each, state two things wrong with the proof: (i) We will prove that AB and BA have the same characteristic equation. We have that det(AB − λI) = det(ABAA−1 − λAA−1) = det(A(BA − λI)A−1) = det(A) + det(BA − λI) − det(A) = det(BA − λI) Hence det(AB − λI) = det(BA − λI), and so...
Select all statements below which are true for all invertible n×n matrices A and B A....
Select all statements below which are true for all invertible n×n matrices A and B A. AB=BA B. (A+B)^2=A^2+B^2+2AB C. (In−A)(In+A)=In−A^2 D. 7A is invertible E. (AB)^−1=A^−1*B^−1 F. A+A^−1 is invertible
Assume A and B are two nonsingular square matrices. Prove that AB has the same eigenvalue...
Assume A and B are two nonsingular square matrices. Prove that AB has the same eigenvalue as BA.
Let matrices A,B∈Mn×n(R). Show that if A and B are each similar to some diagonal matrix,...
Let matrices A,B∈Mn×n(R). Show that if A and B are each similar to some diagonal matrix, and also have the same eigenvectors (but not necessarily the same eigenvalues), then  AB=BA.
If I prove Det(A)Det(B) = Det(AB) for matrices A and B when A and B are...
If I prove Det(A)Det(B) = Det(AB) for matrices A and B when A and B are 2x2 matrices, can I use that to show that Det(A)Det(B) = Det(AB) for any n x n matrix? If so how?
For an n×n matrix, A, the trace of A is defined as the sum of the...
For an n×n matrix, A, the trace of A is defined as the sum of the entries on the main diagonal. That is, tr(A)=a11+a22+?+ann. (a) Prove that for any matrices A and B having the same size, tr(A+B)=tr(A)+tr(B) and for any scalar c, tr(cA)=ctr(A) (b) Prove tr(A)=tr(AT) for all square matrices A. (c) Prove that for any matrices A and B having the same size, tr(AB)=tr(BA). (d) Using (c), prove that if A and B are similar tr(A)=tr(B).
Suppose A, B, C are n x n matrices with det. A =1,det B =-1, det...
Suppose A, B, C are n x n matrices with det. A =1,det B =-1, det C is 2, find AB, A+B,
A and B are two m*n matrices. a. Show that B is invertible. b. Show that...
A and B are two m*n matrices. a. Show that B is invertible. b. Show that Nullsp(A)=Nullsp(BA)