Question

The following system of equations: $$ x_1 + x_2 + x_3 = 1, \quad x_1 + 2x_2 + 3x_3 = 2, \quad x_1 + 4x_2 + \alpha x_3 = 4 $$ has a unique solution. Possible value(s) for $\alpha$ is/are: 

• A. 0
• B. Either 0 or 1
• C. One of 0, 1, or -1
• D. Any real number other than 5

Hint:

For unique solutions, ensure the determinant of the coefficient matrix is non-zero.

Answer 1

The Correct Option is D

To determine the unique solution, calculate the determinant of the coefficient matrix: det = α − 8. For a unique solution, the determinant must not be zero: α − 8 ≠ 0. Thus, α can be any real number except 8.