Question

Which of the following subsets of ℝ3×3 are subspaces of ℝ3×3? A. The 3×3 matrices of rank 2 B. The 3×3 matrices in reduced row-echelon form C. The 3×3 matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries) D. The diagonal 3×3 matrices E. The 3×3 matrices with all zeros in the second row F. The invertible 3×3 matrices

Answer 1

The answer is "Option C, Option D, and Option E".

Explanation:

In point, A: Its matrix (aA+bB) for A, B matrices of rank 2 is not always of rank 2. for Rank(A+B) is smaller or equivalent to RankA+B.

In point B: The two matrices, A, B, and each entry is equal to or larger than zero. Then, take a = -1 = b, and so all entries (aA+bB) belong to the subset are less than or equal to Zero.hence.

In point C: We get the matrix (aA+bB) for A, B matrices with trace 0. Trace zero. (aA + bB) trace is (a×trace(A) + b×trace(B)) thus null.

In point D: For Both A and B, the matrix (aA+bB) is again diagonal for two diagonal matrices.

In point E: For Both A and B matrices, each insertion in second row zero will also have the second-row matrix (aA+bB) as zero.

In point F: In this case, A and B must become two reduced raw echelons, so that A+E is also a reduced raw shape.