linear algebra - Prove that if $V$ = $R^{n,n}$, then the set of all diagonal matrices is a subspace of $V$.

I am reading the book, Applied Linear Algebra and Matrix Analysis.
When I was doing the exercise of Section3.2 Exercise 19, I was puzzled at some of it. Here is the problem description:

Prove that if $V$ = $R^{n,n}$, then the set of all diagonal matrices is a
subspace of $V$.

And I know it is not hard to know the set of all diagonal matrices is closed under matrix addition and scalar multiplication.
BUT, it confused me how to know it contains the zero element of V
SO I check the reference answer which is as followed:

Let A and B be $n×n$ diagonal matrices. Then $cA$ is a diagonal matrix and $A$
$+ B$ is diagonal matrix so the set of diagonal matrices is closed under matrix addition and scalar multiplication.

It doesn't explain anything about zero elements.
And the diagonal matrix is a matrix in which the entries outside the main diagonal are all zero, which means it is can't be equal to zero matrices.
And I wonder the definition of subspace is the only way to prove it in an abstract subspace situation.
If not mind, could anyone help me and give me some inspirations?
Thanks in advance.

Answer

In general, a set $V\subset X$ is a subspace if it is

1. closed under addition


2. closed under scalar multiplication.


3. non-empty.

There is no need to demand that $0\in V$, because that is a consequence of the three properties above, since, if $x\in V$, then $0=x+(-1)\cdot x\in V$.

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Very often, as is your case, the third property, i.e. that $V$ is non-empty, is obvious, and left out of the proof. Technically, it's better to at least mention that of course, $V$ is not empty.

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Just a final remark:

And the diagonal matrix is a matrix in which the entries outside the main diagonal are all zero, which means it is can't be equal to zero matrices.

I don't understand this sentence at all.