The set of all matrices whose entries sum to zero is a subset of
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 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.
denotes the entries in the matrix A
denotes the entries in the matrix B
Consider the sum of entries in A+B
By using summation property,
0 +0 =0
All entries in A+B has sum zero.
A+B lies in W.
Now take any scalar u in real number.
Let A is in W.
We have to show that uA is in W.
Take the sum of all entries in uA
Thus, all entries in uA have sum zero.
Thus, uA lies in W.
Thus W is a subspace of
Therefore, the set of all matrices whose entries sum to zero is a subspace of .
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