2. Let →b Rm, A Mm×n(R) and let R be any row echelon form of A. Which of the following statements must be true?
(a) If the linear system of equations A→x = →b has infinitely many solutions, then R has at least one row of zeros.
(b) If R has at least one row of zeros and the linear system of equations A→x = →b is consistent, then the system has infinitely many solutions.
(c) If the linear system of equations A→x = →b is inconsistent, then R has at least one row of zeros.
(d) The linear systems of equations A→x = →b and R→x = →b have the same set of solutions.
(e) None of the above.
3. Which of the following statements are true for any subspace S of Rn?
(a) If dim(S) = k, then every spanning set for S contains exactly k vectors.
(b) If dim(S) = k, then every linearly dependent set of vectors in S contains more than k vectors.
(c) S is closed under linear combinations.
(d) If S /= {→0}, then there are infinitely many different bases for S.
(e) None of the above.
4. Which of the following statements are true for any A ∈ Mm×n(R)?
(a) The rank of A is the number of nonzero rows in any row echelon form of A.
(b) If B Mm×n(R) can be obtained by performing an elementary row operation on
A, then rank (A) = rank (B).
(c) If A→x = →b is consistent for some →b ∈ Rm, then rank (A) = m.
(d) For any →b ∈ Rm, rank [ A | →b ] ≥ rank (A).
(e) None of the above.
5. Which of the following statements are true for any square matrix A ∈ Mn×n(R)?
(a) If Col (A) = Row (A), then A is symmetric.
(b) If A is symmetric, then Col (A) = Row (A).
(c) If Null (A) = {→0}, then Col (A) = Rn
(d) →0 ∈ Col (A) ∩ Row (A).
(e) None of the above.
6. Which of the following statements hold for any two invertible matrices A, B ∈ Mn×n(R)? (
a) (A + B)−1 = A−1 + B−1.
(b) (AB)−1 = A−1B−1.
(c) A−1 + B−1 = B−1 + A−1. (d) A−1B−1 = B−1A−1.
(e) None of the above.
7. Consider the following directed graph on the three vertices V1, V2, and V3.
How many distinct 4−edged paths are there from V1 to V2.
(a) 2.
(b) 6.
(c) 16.
(d) 4.
(e) None of the above.
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