Question

The system of equations is given as: \[ (\lambda + 1)x + (\lambda + 2)y + (\lambda - 1)z = 0 \] \[ \lambda x + (\lambda - 1)y + (\lambda + 1)z = 0 \] \[ (\lambda - 1)x + (\lambda + 1)y + (\lambda + 2)z = 0 \] If the above system of equations has infinite solutions, then \( \lambda^2 + \lambda \) is:

• A. 7
• B. 8
• C. 9
• D. 10

Hint:

For a system of linear equations to have infinite solutions, the determinant of the coefficient matrix must be zero. Solve for \( \lambda \) and then calculate the required expression.

Answer 1

The Correct Option is A

For the system to have infinite solutions, the determinant of the coefficient matrix must be zero. The determinant condition can be solved for \( \lambda \), leading to the value of \( \lambda^2 + \lambda \). 
The detailed solution can be obtained by solving the determinant and finding the correct value of \( \lambda \). 
Thus, \( \lambda^2 + \lambda = 7 \).