Question

Which of the following subsets of \(\mathbb{R}^{3 \times 3}\) are subspaces of \(\mathbb{R}^{3 \times 3}\)? A. The 3 \(\times\) 3 matrices of rank 2 B. The 3 \(\times\) 3 matrices whose entries are all integers C. The invertible 3 \(\times\) 3 matrices D. The diagonal 3 \(\times\) 3 matrices E. The 3 \(\times\) 3 matrices A such that the vector \(\begin{pmatrix} 3\\4\\5 \end{pmatrix}\) is in the kernel of A F. The 3 \(\times\) 3 matrices with all zeros in the second row

          Which of the following subsets of \(\mathbb{R}^{3 \times 3}\) are subspaces of \(\mathbb{R}^{3 \times 3}\)?
A. The 3 \(\times\) 3 matrices of rank 2
B. The 3 \(\times\) 3 matrices whose entries are all integers
C. The invertible 3 \(\times\) 3 matrices
D. The diagonal 3 \(\times\) 3 matrices
E. The 3 \(\times\) 3 matrices A such that the vector \(\begin{pmatrix} 3\\4\\5 \end{pmatrix}\) is in the kernel of A
F. The 3 \(\times\) 3 matrices with all zeros in the second row

Which of the following subsets of ℝ^3 × 3 are subspaces of ℝ^3 × 3?
A. The 3 × 3 matrices of rank 2
B. The 3 × 3 matrices whose entries are all integers
C. The invertible 3 × 3 matrices
D. The diagonal 3 × 3 matrices
E. The 3 × 3 matrices A such that the vector < p m a t r i x > is in the kernel of A
F. The 3 × 3 matrices with all zeros in the second row

Added by Kevin C.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

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Solved by Expert Rylie Howey

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- A subset \( S \) of \( \mathbb{R}^{3 \times 3} \) is a subspace if it satisfies three conditions: it contains the zero matrix, it is closed under addition (if \( A, B \in S \), then \( A + B \in S \)), and it is closed under scalar multiplication (if \( A \in S Show more…

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Which of the following subsets of R3X3 are subspaces of R³×3³? A. The 3 x 3 matrices of rank 2 B. The 3 x 3 matrices whose entries are all integers C. The invertible 3 x 3 matrices D. The diagonal 3 x 3 matrices () 4 5 OF. The 3 x 3 matrices with all zeros in the second row E. The 3 x 3 matrices A such that the vector is in the kernel of A Which of the following subsets of IR3x3 are subspaces of IR3x3? A.The 3x3matrices of rank 2 B.The 3 X 3 matrices whose entries are all integers c.The invertible 3x3 matrices D.The diagonal 3 3 matrices 3 E.The 3X 3 matrices Asuch that the vector 4 is in the kernel of A F.The 3X 3 matrices with all zeros in the second row