Previous Problem List Next (1 point) Let R3I be the vector space of 3x3 matrices

Question

Previous Problem List Next (1 point) Let R3I be the vector space of 3x3 matrices. Which of the following subsets of RSI are subspaces of R33 A. The non-invertible 3x3 matrices B. The 3 x 3 matrices whose entries are all greater than or equal to 0 (0) (0) c. The 3 x 3 matrices A such that A1 = 0 ) D. The 3 x 3 matrices that, after being transformed to row-echelon form, have 2 nonzero rows E. The diagonal 3 x 3 matrices EThe 3 x 3 matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries)

Explanation / Answer

A. Not a subspace.

Consider two matrices in this set.

X,Y,both diagonal matrices

X has entry =1 at (1,1) and (2,2) and Y has entry equal to 1 at (3,3)

X+Y =I which is invertible

B. Not a subspace

Take any element in this set and multiply by -1 and the resulting matrix is not in the set

C. This is a subspace

Denote :[0,1,8]=b

Let, X ,Y be in the set and c ascalar

(X+Y)b=Xb+Yb=0

(cX)b=cXb=0

D. Not a subspace

Let, X be in the set

X+(-1)X=0 which is not in the set

E. It is a subspace

F. It is a subspace

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