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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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

Isma Jimenez · Accepted Answer

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  u∑i,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.

Jeffrey Jordon · Accepted Answer

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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

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