Question

The system of linear equations
\(x + y + z = 2\)
\(2x + 3y + 2z = 5\)
\(2x + 3y + (a^2 - 1)z = a + 1\)
then

• A. has infinitely many solutions for \(a = 4\)
• B. has a unique solution for \(|a| = \sqrt3\)
• C. is inconsistent when \(|a| = \sqrt3\)
• D. is inconsistent when \(a = 4\)

Hint:

Eliminate by row operations; the last equation reduces to \((a^2 - 3)z = a - 4\).

Answer 1

The Correct Option is C

\(R_2 - 2R_1\): \(y = 1\). \(R_3 - 2R_1\): \(y + (a^2 - 3)z = a - 3\). Subtract \(y = 1\): \((a^2 - 3)z = a - 4\).
If \(a^2 \ne 3\): unique solution \(z = \frac{a - 4}{a^2 - 3}\), then \(x = 1 - z\). For \(a = 4\): \(z = 0\), \(x = 1\), \(y = 1\), a unique solution (not infinite, not inconsistent).
If \(|a| = \sqrt3\): the equation becomes \(0 \cdot z = a - 4\) with \(a - 4 \ne 0\) (as \(a = \pm\sqrt3\)), which is impossible, so the system is inconsistent.
Only option (3) is true.
Answer: (3) is inconsistent when \(|a| = \sqrt3\)