Let $T$ be a $4 \times 4$ matrix with real entries Let $T$ be $4\times 4$ matrix with real entries. Suppose $T^5=0$. Then which of the following is necessarily true?
(A) $T$ is the zero matrix.
(B) $T$ need not be the zero matrix, but $T^2$ is the zero matrix.
(C) $T^2$ need not be the zero matrix, but $T^3$ is the zero matrix.
(D) $T^3$ need not be the zero matrix, but $T^4$ is the zero matrix.  
How can I tackle the above problem? Any help will be appreciated.Thanks in advance for your time.
 A: A nilpotent matrix's degree should be less that or equal to its dimension, so your matrix's degree is less or equal to 4 so :
statement (D) is correct and all other statements are false, because of this example: 
$N$ = \begin{bmatrix} 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end{bmatrix} 
$N^2$ = \begin{bmatrix} 0 & 0 & 2 & 7\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} $N^3$ = \begin{bmatrix} 0 & 0 & 0 & 6\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} $N^4$ = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} $N^5$ = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
So $T$ need not to be zero then statement (A) is false
$T^2$ need not be zero then statement (B) is false
$T^3$ need not be zero then statement (c) is false
every $4\times 4$ matrix that is nilpotent should have a degree less than or equal to $4$, $T$ is nilpotent, then $T$'s degree is less than or equal to $4$, then $T^4$ need to be zero, then statement(D) is true
ATTENTION:
for disproving a statement, a negative example suffices, so for disproving statements (A), (B) and (C) the example works
But for proving statement (D) you should first prove this theorem:
` A is a n×n matrix, Am=0 for some positive integer m. Show that An=0.`

then you could see that (D) is true.
