Question

Determine the value of $\lambda$ and $\mu$ for which such that the system of equations $x + 2y + z = 6$, $x + 4y + 3z = 10$, and $2x + 4y + \lambda z = \mu$ has infinite number of solutions.

• A. $\lambda = 2, \mu \neq 12$
• B. $\lambda = 2, \mu = 12$
• C. $\lambda \neq 2, \mu = 12$
• D. for any $\lambda$ and any $\mu$

Hint:

For systems of linear equations to have infinite solutions, the determinant of the coefficient matrix must be zero, and the right-hand side must satisfy the corresponding conditions.

Answer 1

The Correct Option is C

To determine when the system has infinite solutions, the determinant of the coefficient matrix must be zero. We have the system of equations: \[ \begin{aligned} x + 2y + z &= 6 \\ x + 4y + 3z &= 10 \\ 2x + 4y + \lambda z &= \mu \end{aligned} \] The coefficient matrix for the system is: \[ \begin{pmatrix} 1 & 2 & 1 \\ 1 & 4 & 3 \\ 2 & 4 & \lambda \end{pmatrix} \] For the system to have infinite solutions, the determinant of the matrix must be zero: \[ \text{Determinant} = \begin{vmatrix} 1 & 2 & 1 \\ 1 & 4 & 3 \\ 2 & 4 & \lambda \end{vmatrix} = 0 \] Calculating the determinant: \[ = 1\begin{vmatrix} 4 & 3 \\ 4 & \lambda \end{vmatrix} - 2\begin{vmatrix} 1 & 3 \\ 2 & \lambda \end{vmatrix} + 1\begin{vmatrix} 1 & 4 \\ 2 & 4 \end{vmatrix} \] \[ = 1(4\lambda - 12) - 2(\lambda - 6) + 1(4 - 8) \] \[ = 4\lambda - 12 - 2\lambda + 12 - 4 = 2\lambda - 4 \] For infinite solutions, we set the determinant to zero: \[ 2\lambda - 4 = 0 \Rightarrow \lambda = 2 \] Now, substitute \( \lambda = 2 \) into the third equation: \[ 2x + 4y + 2z = \mu \] For the system to have infinite solutions, the third equation must be a linear combination of the first two. By subtracting the first from the second, we get: \[ (x + 4y + 3z) - (x + 2y + z) = 2y + 2z = 4 \Rightarrow y + z = 2 \] We also need the third equation to be consistent with this. Using \( 2x + 4y + 2z = \mu \), substitute from equations 1 and 2 or reduce them, and we find: \[ \mu = 12 \] Thus, the correct condition for infinite solutions is: \( \lambda = 2 \) and \( \mu = 12 \).