Prove or disprove the statement:
If all the eigenvalues of a matrix are $0$, then the matrix must be the zero matrix.
What I know : If the matrix is a upper or lower triangle matrix with the diagonal entries all being zero then the matrix should not necessarily be a zero matrix.
Can anyone direct me along the a way to prove this please ?
The proof is equivalent to giving a counterexample of the statement. Note that this satisfies the logical properties of a proof. Because indeed it proves that the negation of the statement is true. To this end, see if you can form the logical negation of the statement (have you tried writing this statement mathematically, even?):
The following is NOT true $$\forall \text{matrices $M$ with only zero eigenvalues, $M$ is the zero matrix}$$
If and only if $$\exists \text{matrix $M_0$ such that $M_0$ has zero eigenvalues AND $M_0$ is not the zero matrix.}$$ If you can find just a single $M_0$ that satisfies the above, you are done, and you have "disproved" the statement.