The following are true or false. If true, provide an explanation why it is true, and if it is false provide an example demonstrating this along with an explanation why it shows the statement is incorrect.
(a) If
A
is strictly diagonally dominant, then
A
T
is strictly diagonally dominant.
(b) If
A
is strictly diagonally dominant, then αA, where α is a nonzero number, is strictly diagonally dominant.
(c) If A is symmetric, then ||A||
= ||A||1.
(d) If a nonzero vector
x
can be found so
Ax
= 0, where
A
is symmetric, then
A
is not positive definite.
(e) If
A
is positive definite, and symmetric, then
A
only has positive entries.
(f) If
A
is the 2 × 2 zero matrix and
A
=
LU, then either
L
or
U
is the zero matrix.
(g) Because ||x|| ≤ ||x||1
then it must be that ||A|| ≤ ||A||1.
(h) Assuming
A
is 2 × 2, if
A
is symmetric and positive definite, then
A
−1
is symmetric and positive definite.
(i) A symmetric and positive definite matrix must be strictly diagonally dominant.
(j) An ill-conditioned matrix can be made well conditioned using pivoting (you can assume that the matrix is 2 × 2).