Question

Let \( R \) be the row reduced echelon form of a \( 4 \times 4 \) real matrix \( A \) and let the third column of \( R \) be \[ \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}. \] Consider the following statements: \[ P: \text{If} \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} \text{ is a solution of } A x = 0, \text{ then } \gamma = 0. \] \[ Q: \text{For all } b \in \mathbb{R}^4, \text{rank}[A | b] = \text{rank}[R | b]. \] Then: \\

• A. both P and Q are TRUE
• B. P is TRUE and Q is FALSE
• C. P is FALSE and Q is TRUE
• D. both P and Q are FALSE

Hint:

In a row reduced echelon form, the solution to the system depends on the pivot positions and the consistency of the system, not just the form of the matrix.

Answer 1

The Correct Option is D

- Statement P: From the given row reduced echelon form, we know that the third column has a leading 1 in the second row, which implies that \( \beta \) is a free variable while \( \gamma \) must be zero. Thus, the statement is FALSE because \( \gamma \) should not necessarily be zero in every case. - Statement Q: The rank of the augmented matrix \( [A | b] \) and the row reduced matrix \( [R | b] \) can be different when there is a row in the augmented matrix that is inconsistent. Therefore, the statement is FALSE. Final Answer: (D) both P and Q are FALSE