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

Infinite solutions exist for the system if the determinant of the coefficient matrix is zero. Solving the determinant condition for \( \lambda \) yields \( \lambda^2 + \lambda \). The specific value of \( \lambda \) can be determined by solving the determinant. Consequently, \( \lambda^2 + \lambda = 7 \).