Question # Determine whether the subset of M_{n,n} is a subspace of M_{n,n} with the standard operations. Justify your answer. The set of all n times n matrices whose entries sum to zero

Matrices
ANSWERED Determine whether the subset of $$M_{n,n}$$ is a subspace of $$M_{n,n}$$ with the standard operations. Justify your answer.
The set of all $$n \times n$$ matrices whose entries sum to zero 2020-11-11
Step 1
Given that,
The set of all $$n \times n$$ matrices whose entries sum to zero is a subset of $$M_{n,n}$$
A nonempty subset W of vector space V is a subspace if it is closed under addition and scalar multiplication.
That is, if u, v in W then u +v lies in W.
If a is any scalar then au also in W.
Let W is the set of all $$n \times n$$ matrices whose entries sum to zero.
As n by n zero matrix whose entries sum to zero.
Thus W is non-empty.
Let A and B are two n by n matrix such that all entire sum add up to zero.
$$a_{ij},\dotsc,i,j=1,2,3,\dotsc n$$ denotes the entries in the matrix A
$$b_{ij},\dotsc,i,j=1,2,3,\dotsc n$$ denotes the entries in the matrix B
Thus, $$\sum_{i,j=1}^n a_{ij}=0$$
$$\sum_{i,j=1}^n b_{ij}=0$$
Step 2
Consider the sum of entries in A+B
$$\sum_{i,j=1}^n a_{ij}+b_{ij}$$
By using summation property,
$$\sum_{i,j=1}^n a_{ij}+\sum_{i,j=1}^n b_{ij}$$
It gives,
0 +0 =0
Thus, $$\sum_{i,j=1}^n a_{ij}+ b_{ij}=0$$
Therefore,
All entries in A+B has sum zero.
A+B lies in W.
Step 3
Now take any scalar u in real number.
Let A is in W.
$$\sum_{i,j=1}^n a_{ij}=0$$
We have to show that uA is in W.
Take the sum of all entries in uA
$$\sum_{i,j=1}^n ua_{ij}$$
$$u\sum_{i,j=1}^n a_{ij}$$
$$u(0)=0$$
Thus, all entries in uA have sum zero.
Thus, uA lies in W.
Thus W is a subspace of $$M_{n,n}$$
Therefore, the set of all $$n \times n$$ matrices whose entries sum to zero is a subspace of $$M_{n,n}$$.