Question

The system of equations

–kx + 3y – 14z = 25

–15x + 4y – kz = 3

–4x + y + 3z = 4

is consistent for all k in the set

• A. R
• B. R – {–11, 13}
• C. R – {13}
• D. R – {–11, 11}

Answer 1

The Correct Option is A

 To determine the consistency of the given system of equations for all values of \( k \), we need to examine the system:

1. \(-kx + 3y - 14z = 25\)
2. \(-15x + 4y - kz = 3\)
3. \(-4x + y + 3z = 4\)

A system of linear equations is consistent if there is at least one solution. This can be analyzed using the determinant of the coefficient matrix. If the determinant is non-zero, the system is consistent for all values of parameters; here, parameters are \( x \), \( y \), and \( z \).

The coefficient matrix of the system is:

\(-k\)	\(3\)	\(-14\)
\(-15\)	\(4\)	\(-k\)
\(-4\)	\(1\)	\(3\)

The determinant of this matrix is:

\(\Delta = \begin{vmatrix} -k & 3 & -14 \\ -15 & 4 & -k \\ -4 & 1 & 3 \end{vmatrix}\)

Expand by the first row:

\(-k \left( \begin{vmatrix} 4 & -k \\ 1 & 3 \end{vmatrix} \right) - 3 \left( \begin{vmatrix} -15 & -k \\ -4 & 3 \end{vmatrix} \right) - 14 \left( \begin{vmatrix} -15 & 4 \\ -4 & 1 \end{vmatrix} \right)\)

Calculate these minors:

\(\begin{vmatrix} 4 & -k \\ 1 & 3 \end{vmatrix} = (4 \cdot 3) - (-k \cdot 1) = 12 + k = 12 + k\)

\(\begin{vmatrix} -15 & -k \\ -4 & 3 \end{vmatrix} = ((-15) \cdot 3) - ((-k) \cdot (-4)) = -45 - 4k\)

\(\begin{vmatrix} -15 & 4 \\ -4 & 1 \end{vmatrix} = ((-15) \cdot 1) - (4 \cdot (-4)) = -15 + 16 = 1\)

Substitute these into the determinant:

\(\Delta = -k(12 + k) + 3(-45 - 4k) - 14(1)\)

Expand and simplify:

\(=-k^2 - 12k - 135 - 12k - 14 = -k^2 - 24k - 149\)

Given the task is to ensure consistency for all \( k \), for the determinant \( \Delta \) does not need conditions to be non-zero unless any value of \( k \) causes the system to be inconsistent, which in this case does not occur. Thus, the system is consistent for all \( k \).

Therefore, the correct answer is the set \(R\), meaning the system is consistent for all real numbers \( k \).