Question

A, C are \( 3 \times 3 \) matrices. B, D are \( 3 \times 1 \) matrices. If \( AX=B \) has a unique solution and \( CX=D \) has an infinite number of solutions, then

• A. rank of \( [A:D] = \text{rank of } [C:B] \)
• B. rank of \( A = \text{rank of } C \)
• C. rank of \( [A:B]<\text{rank of } [B:D] \)
• D. rank of \( [A:D] \ge \text{rank of } [C:B] \)

Hint:

Always recall: A unique solution for an \( n \times n \) system implies the coefficient matrix has rank \( n \). Infinite solutions imply rank \(<n \). The rank of an augmented matrix cannot exceed the number of rows.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We use the properties of the rank of matrices in systems of linear equations (Rouche-Capelli Theorem). For a system \( MX = Y \) with \( M \) being \( 3 \times 3 \): 1. Unique Solution: \( \text{rank}(M) = \text{rank}[M:Y] = 3 \) (Full rank). 2. Infinite Solutions: \( \text{rank}(M) = \text{rank}[M:Y] < 3 \).
Step 2: Analyzing the Data:
* System \( AX = B \): Has a unique solution. Since \( A \) is \( 3 \times 3 \), this implies \( \text{rank}(A) = 3 \). Consequently, the augmented matrix \( [A:D] \) (which is \( 3 \times 4 \)) must have a rank of 3 because it contains a \( 3 \times 3 \) submatrix \( A \) with rank 3, and the maximum possible rank for a matrix with 3 rows is 3. So, \( \text{rank}[A:D] = 3 \). * System \( CX = D \): Has infinite solutions. Since \( C \) is \( 3 \times 3 \), this implies \( \text{rank}(C) < 3 \). Also, \( \text{rank}(C) = \text{rank}[C:D] = k \), where \( k \in \{1, 2\} \).
Step 3: Evaluating Options:
Now we evaluate \( \text{rank}[C:B] \). This is an augmented matrix of size \( 3 \times 4 \). Its rank can be at most 3 (limited by the number of rows). So, \( \text{rank}[C:B] \le 3 \). Comparison: We established that \( \text{rank}[A:D] = 3 \). We know that \( \text{rank}[C:B] \le 3 \). Therefore, \( \text{rank}[A:D] \ge \text{rank}[C:B] \) is always true (since \( 3 \ge \text{anything} \le 3 \)). Let's check other options:

• (A) \( \text{rank}[A:D] = \text{rank}[C:B] \): Not necessarily true. \( [C:B] \) could have rank 2.
• (B) \( \text{rank}(A) = \text{rank}(C) \): False. \( 3 \neq < 3 \).
• (C) \( \text{rank}[A:B] < \text{rank}[B:D] \): \( \text{rank}[A:B]=3 \). Rank of \( [B:D] \) is max 3 (as it has 3 rows or assuming it's formed by columns, wait \( B, D \) are \( 3 \times 1 \), so \( [B:D] \) is \( 3 \times 2 \), max rank 2). So \( 3 < 2 \) is False.

Step 3: Final Answer:

Option (D) is the correct statement.