Determine if the statement is true or false and

Determine if the statement is true or false, and justify your answer. You may assume that A and B are
matrices.
If then either or
n
×
n
A − B .
True. If i
≠
j , then A
ij
= B
ij
= 0, so ( A − B )
ij
= 0, and A − B is diagonal.
True. If i = j , then A
ij
= B
ij
= 0, so ( A − B )
ij
= 0, and A − B is diagonal.
False. If i = j , then ( A − B )
ij
= A
ij
− B
ij
may not be zero, so A − B is not necessarily diagonal.
False. If i
≠
j , then ( A − B )
ij
= A
ij
− B
ij
may not be zero, so A − B is not necessarily diagonal.
False. If i
≠
j , then A
ij
= − B
ij
, so ( A − B )
ij
= 2 A
ij
, and A − B is not diagonal.
n
×
n
AB = BA , A = I
n
B = I
n
.
True. AB = BA can only be true if either A = I
n
or B = I
n
.
False. For example, A = B = 0
nn
.
False. For example, A = , B = .
01
10
−10
01
False. AB = BA for any matrices A
≠
B .
False. For example, A = , B = .
01
10
10
01