Question

If the system of equations \[ (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \] \[ \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \] \[ (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \] has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to:

• A. 10
• B. 12
• C. 6
• D. 20

Hint:

To find the value of \( \lambda \) for infinitely many solutions in a system of linear equations, set the determinant of the coefficient matrix to zero and solve for \( \lambda \).

Answer 1

The Correct Option is B

Determinant Calculation for Infinitely Many Solutions

For infinitely many solutions, the determinant of the coefficient matrix must be zero:

D = \( \begin{vmatrix} \lambda - 1 & \lambda - 4 & \lambda \\ \lambda & \lambda - 1 & \lambda - 4 \\ \lambda + 1 & \lambda + 2 & - (\lambda + 2) \end{vmatrix} \) = 0

Expanding the determinant:

(\(\lambda - 3\)(2\(\lambda\) + 1)) = 0

This gives us:

\(\lambda = 3\) or \(\lambda = -\frac{1}{2}\)

Next, we find \(\lambda^2 + \lambda\).

For \(\lambda = 3\):

\(\lambda^2 + \lambda = 3^2 + 3 = 9 + 3 = 12\)

Thus, the correct answer is:

\(\lambda^2 + \lambda = 12\)