Let $T$ be a $4 \times 4$ matrix with real entries

Question

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.

Do you know of the minimal polynomial of a matrix, or the Cayley-Hamilton theorem? — Andrés E. Caicedo, Jan 20 '13 at 06:41
The answer is here: http://en.wikipedia.org/wiki/Nilpotent_matrix. Look for the line: "The degree of an $n\times n$ nilpotent matrix is always less than or equal to $n$," and the phrase "canonical nilpotent matrix." The proof is another story. — Anon, Jan 20 '13 at 06:44
Thanks for the link.It's been useful. Now degree of $4 \times 4$ nilpotent matrix $T$ must be $\leq 4 $. From here i can not decide about which of the given options can be correct.Can you help me a little bit more? — , Jan 20 '13 at 06:55
@user53386: How does that canonical nilpotent matrix factor in? — Jonas Meyer, Jan 20 '13 at 06:57
@JonasMeyer sir,pardon me for my ignorance but i do not have any idea about Canonical nilpotent matrix even though i know about nilpotent matrix. So,if you explain a little bit more in this regard,i can progress. — , Jan 20 '13 at 07:03
@user3386: You may find what a canonical nilpotent matrix is on the link that I have mentioned. To help you a little more, consider the following problem instead: Let $T$ be a $2 \times 2$ matrix with real entries. Suppose $T^3 = 0$. Which of the following is true: A) $T$ is the all-zero matrix, B) $T$ need not be all-zero, but $T^2$ is all-zero. — Anon, Jan 20 '13 at 07:05
@Anon I have got your point sir. Surely $(B)$ will be the correct choice. So here in the aforementioned problem ,i guess $(D)$ is the right choice.Am i right sir? — , Jan 20 '13 at 07:08
@user53386: anon had mentioned that the phrase "canonical nilpotent matrix" can be found on the Wikipedia page in a prior comment. Ctrl F will help you find it. — Jonas Meyer, Jan 20 '13 at 07:11
@JonasMeyer thanks a lot sir.I have got it using Ctrl $F$.Earlier i did not find that. — , Jan 20 '13 at 07:14
@user53386 Yes, you are right. I suggest you make sure you construct a matrix $T$ such that it satisfies the statement of the question, but does not satisfy A, B, and C. Then, you need to show that D holds. A good place to start is the link provided by Jonas Meyer. — Anon, Jan 20 '13 at 07:16

score 3 · Accepted Answer · edited Apr 15 '13 at 14:29

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.

Thanks a lot sir for the detailed discussion of the problem .I have got it now. — , Jan 22 '13 at 04:28

Let $T$ be a $4 \times 4$ matrix with real entries

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