Question

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

Answer #1

Linear Algebra question:Suppose A and B are invertible
matrices,with A being m*m and B n*n.For any m*n matrix C and any
n*m matrix D,show that:
a)(A+CBD)-1-A-1C(B-1+
DA-1C)-1DA-1
b) If A,B and A+B are all m*m invertible matrices,then deduce
from a) above that
(A+B)-1=A-1-A-1(B-1+A-1)-1A-1

Suppose A and B are invertible matrices, with A being m x m and
B being n x n. For any m x n matrix C and any n x m matrix D, show
that :
a) (A + CBD)-1 = A-1-
A-1C(B-1 +
DA-1C)-1DA-1
b) If A, B and A + B are all m x m invertible matrices, then
deduce from (a) above that (A + B)-1 = A-1 -
A-1(B-1 +
A-1)-1A-1

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

We say two n × n matrices A and B are similar if there is an
invertible n × n matrix P such that
A = PBP^ -1.
a) Show that if A and B are similar n × n matrices, then they
must have the same determinant.
b) Show that if A and B are similar n × n matrices, then they
must have the same eigenvalues.
c) Give an example to show that A and B can be...

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).

Suppose A and B are invertible matrices in Mn(R) and
that A + B is also invertible. Show that C = A-1 +
B-1 is also invertible.

Let A, B ? Mn×n be invertible matrices. Prove the following
statement: Matrix A is similar to B if and only if there exist
matrices X, Y ? Mn×n so that A = XY and B = Y X.

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.

Show that the product of two n × n unitary matrices is unitary.
Is the same true of the sum of two n × n unitary matrices? Prove or
find a counterexample.

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.

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