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

CoormaBak9 2020-11-10 Answered
Determine whether the subset of Mn,n is a subspace of Mn,n with the standard operations. Justify your answer.
The set of all n×n matrices whose entries sum to zero
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Expert Answer

Isma Jimenez
Answered 2020-11-11 Author has 84 answers
Step 1
Given that,
The set of all n×n matrices whose entries sum to zero is a subset of Mn,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×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.
aij,,i,j=1,2,3,n denotes the entries in the matrix A
bij,,i,j=1,2,3,n denotes the entries in the matrix B
Thus, i,j=1naij=0
i,j=1nbij=0
Step 2
Consider the sum of entries in A+B
i,j=1naij+bij
By using summation property,
i,j=1naij+i,j=1nbij
It gives,
0 +0 =0
Thus, i,j=1naij+bij=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.
i,j=1naij=0
We have to show that uA is in W.
Take the sum of all entries in uA
i,j=1nuaij
ui,j=1naij
u(0)=0
Thus, all entries in uA have sum zero.
Thus, uA lies in W.
Thus W is a subspace of Mn,n
Therefore, the set of all n×n matrices whose entries sum to zero is a subspace of Mn,n.
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Jeffrey Jordon
Answered 2022-01-30 Author has 2262 answers

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