Let R3 times 3 be the vector space 3 times 3 matrices Which

Let R^3 times 3 be the vector space 3 times 3 matrices. Which of the following subsets of R^3 times 3 are subspaces of R^3 times 3? The invertible 3 times 3 matrices The 3 times 3 matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries) The diagonal 3 times 3 matrices The 3 times 3 matrices whose entries are all greater than or equal to 0 The non-invertible 3 times 3 matrices The 3 times 3 matrices with all zeros in the second row

Solution

A) the invertible 3x3 matrix.

This is not a subspace because the zero matrix is not in this subset.

B) this is a subspace of R^3x3 since, for all vector A, B R^3x3, we have, A+B R^3x3

and the subset is closed under scaler multiplication.

Also, the zero matrix belongs to this subset.

C) This is not a subspace because the zero matrix does not belong to this subset.

D) This is a subspace becuase all the properties of the subspace are satisfied by this subset.

E) This is a subspace since zero matrix belongs to this subset. And, all the properties of subspace are satisfied by this subset.

F) Since all the elements in the second row are zero, the matrices are non-invertible.

Hence, this subset is a subspace.