Question

The values of \( p \) and \( q \) so that the system of equations \[ \begin{aligned} 2x + py + 6z &= 8, \\ x + 2y + qz &= 5, \\ x + y + 3z &= 4 \end{aligned} \] may have no solution are:

• A. \( p \neq 2, \, q = 3 \)
• B. \( p \neq 2, \, q \neq 3 \)
• C. \( p = 2, \, q = \frac{15}{4} \)
• D. \( p = 2, \, q = 3 \)

Hint:

To check for inconsistency in a system of equations, compare the rank of the coefficient matrix and the augmented matrix. Inconsistency occurs when ranks differ.

Answer 1

The Correct Option is A

Step 1: For a system of linear equations to have no solution, the equations must be inconsistent. This generally happens when: \[ \text{Rank of coefficient matrix} \neq \text{Rank of augmented matrix} \] Step 2: Consider the coefficient matrix: \[ A = \begin{bmatrix} 2 & p & 6 \\ 1 & 2 & q \\ 1 & 1 & 3 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 2 & p & 6 & 8 \\ 1 & 2 & q & 5 \\ 1 & 1 & 3 & 4 \end{bmatrix} \] Step 3: We aim to choose \( p \) and \( q \) such that one of the rows in the augmented matrix leads to an inconsistency after row reduction. Testing multiple values, we find that the system becomes inconsistent (no solution) when: \[ p \neq 2, \quad \text{and} \quad q = 3 \] For these values, the last row in the row-reduced form turns into something like: \[ 0x + 0y + 0z = c \quad \text{where} \quad c \neq 0 \] indicating inconsistency.