Question

(7) Prove the following statements.

(c) If A is invertible and similar to B, then B is invertible and A−1 is similar to B−1 .

(d) The trace of a square matrix is the sum of the diagonal entries in A and is denoted by tr A. It can be verified that tr(F G)=tr(GF) for any two n × n matrices F and G. Prove that if A and B are similar, then tr A = tr B

Answer #1

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).

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.

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...

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...

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

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

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).

1. Determine if the statements are true or false:
a. The eigenvalues of a lower triangular matrix are the diagonal
entries of the matrix.
b. For every square matrix A, the sum of all the eigenvalues of
A is equal to the sum of all the diagonal entries of A.

Deside whether the statements below are true or false. If
true, explain why true. If false, give a counterexample.
(a) If a square matrix A has a row of zeros, then A is not
invertible.
(b) If a square matrix A has all 1s down the main diagonal,
then A is invertible.
(c) If A is invertible, then A−1 is invertible.
(d) If AT is invertible, then A is invertible.

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