Part I: True/False For each of the following mark as (T) true or (F) false. If you marked as false, provide a counterexample. If A and B are n times n lower triangular matrices, then AB is also lower triangular. If A and B are n times n symmetric matrices, then AB is symmetric. If A is a 4 times 2 matrix, then Ax=0 has at least two free variables. solutionspile.com

Question

Part I: True/False For each of the following mark as (T) true or (F) false. If you marked as false, provide a counterexample. If

A

and

B

are

n\times n

lower triangular matrices, then

AB

is also lower triangular. If

A

and

B

are

n\times n

symmetric matrices, then

AB

is symmetric. If

A

is a

4\times 2

matrix, then

Ax=0

has at least two free variables.

tr(ABC)=tr(BAC)

for an

n\times n

matrices

A,B

, and

C

. If

W

is a subspace of a vector space

V

and

B

is a basis for

W

, then

B

can be extended to a basis for

V

.

tr(AB)=tr(A)tr(B)

for an

n\times n

matrices

A,B

, and

C

. If

W

is a subspace of a vector space

V

and

B

is a basis for

V

, then

B

can be restricted to a basis for

W

. For any

m\times n

matrices

A

and

B

,

B=EA for some invertible ELongleftrightarrowNS(A)=NS(B).

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