Question

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.

Answer #1

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 ^ 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 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. 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 as BA.

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 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 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 C is 2, find AB, A+B,

A and B are two m*n matrices. a. Show that B is invertible. b.
Show that Nullsp(A)=Nullsp(BA)

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