The subset of matrices in the vector space

Answered question

2022-04-05

The subset of matrices in the vector space 𝑀𝑛,𝑛 of 𝑛 × 𝑛 matrices whose diagonal elements sum to 0.
 

Answer & Explanation

nick1337

nick1337

Expert2022-04-19Added 777 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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