Which of the following subsets of R3 times 3 are subspaces o

Which of the following subsets of R^3 times 3 are subspaces of R^3 times 3? The 3 times 3 matrices whose entries are all greater than or equal to 0 The 3 times 3 matrices in reduced row-echelon form The diagonal 3 times 3 matrices The symmetric 3 times 3 matrices The 3 times 3 matrices with all zeros in the second row The 3 times 3 matrices whose entries are all integers

Solution

The set of all 3 x 3 matrices , all whose entries are greater than 0 is not closed under scalar multiplication ( when the scalar is negative, the resultant matrix will have all negative entries). Therefore, it is not a vector space. The set of all 3 x3 matrices in RREF form is not closed under addition (I3 + I3 is not in its reduced row-echelon form). Therefore, it is not a vector space. The set of 3 x 3 diagonal matrices is closed under addition and scalar multiplication. Further, the       3 x 3 zero matrix is also a diagonal matrix. Hence the set is a vector space. The set of all 3 x3 symmetric matrices is closed under addition and scalar multiplication. Further, the 3 x 3 zero matrix is also a symmetric matrix. Hence the set is a vector space. The set of all 3 x3 s matrices with all zeroes in the 2nd row is closed under addition and scalar multiplication. Further, the 3 x 3 zero matrix is also a member of this set. Hence the set is a vector space. The set of all 3 x 3 matrices , all whose entries are integers is not closed under scalar multiplication      ( when the scalar isnot an integer, the resultant matrix will not have integer entries). Therefore, it is not a vector space.