Question

The system of equations \(3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0\) is

• A. Inconsistent
• B. Has only the trivial solution x = y = z = 0
• C. Reduced to single equation and solution does not exist
• D. Determinant of the coefficient matrix is zero

Hint:

Homogeneous Systems (AX=0). Always consistent (trivial solution x=0 exists). Has only the trivial solution if \(\det(A) \neq 0\). Has non-trivial (infinite) solutions if \(\det(A) = 0\).

Answer 1

The Correct Option is B

This is a system of homogeneous linear equations (AX = 0)
A homogeneous system always has at least one solution, the trivial solution (x=y=z=0)
It has non-trivial solutions if and only if the determinant of the coefficient matrix A is zero (i
e
, the matrix is singular)
Let's find the determinant of the coefficient matrix A: $$ A = \begin{pmatrix} 3 & 2 & 1 \\ 1 & 4 & 1 \\ 2 & 1 & 4 \end{pmatrix} $$ $$ \det(A) = 3(4 \times 4 - 1 \times 1) - 2(1 \times 4 - 1 \times 2) + 1(1 \times 1 - 4 \times 2) $$ $$ \det(A) = 3(16 - 1) - 2(4 - 2) + 1(1 - 8) $$ $$ \det(A) = 3(15) - 2(2) + 1(-7) $$ $$ \det(A) = 45 - 4 - 7 = 34 $$ Since the determinant (\(\det(A) = 34\)) is non-zero, the system of homogeneous equations has only the unique trivial solution, which is x = y = z = 0
Option (4) is incorrect
Option (1) is incorrect as homogeneous systems are always consistent
Option (3) is incorrect
Option (2) is correct