 # 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 ${M}_{n,n}$ is a subspace of ${M}_{n,n}$ with the standard operations. Justify your answer.
The set of all $n×n$ matrices whose entries sum to zero
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Step 1
Given that,
The set of all $n×n$ 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 $n×n$ 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.
${a}_{ij},\dots ,i,j=1,2,3,\dots n$ denotes the entries in the matrix A
${b}_{ij},\dots ,i,j=1,2,3,\dots n$ 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
$\sum _{i,j=1}^{n}u{a}_{ij}$
$u\sum _{i,j=1}^{n}{a}_{ij}$
$u\left(0\right)=0$
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 $n×n$ matrices whose entries sum to zero is a subspace of ${M}_{n,n}$.
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