Question

which of the following statements is true about the rank of a nonzero $5 \times 9$ matrix $a$?\
(a) $\operatorname{rank}(a) \geq 9$\
(b) $1 \leq \operatorname{rank}(a) \leq 9$\
(c) $\operatorname{rank}(a) \geq 5$\
(d) $0 \leq \operatorname{rank}(a) \leq 5$\
(e) $1 \leq \operatorname{rank}(a) \leq 5$

Explanation:

Step1: Recall Rank Properties

The rank of a matrix \( A \) (denoted \( \text{rank}(A) \)) is the maximum number of linearly independent rows (or columns). For an \( m \times n \) matrix, \( \text{rank}(A) \leq \min(m, n) \). Also, for a nonzero matrix, \( \text{rank}(A) \geq 1 \) (since there's at least one nonzero row/column).

Step2: Apply to \( 5 \times 9 \) Matrix

Here, \( m = 5 \), \( n = 9 \). So \( \text{rank}(A) \leq \min(5, 9) = 5 \). Since \( A \) is nonzero, \( \text{rank}(A) \geq 1 \). Thus, \( 1 \leq \text{rank}(A) \leq 5 \).

Answer:

E. \( 1 \leq \text{rank}(A) \leq 5 \)