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
asked 2020-11-10
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

Answers (1)

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 \(M_{n,n}\).
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