Question

If the system of equations: \[ 3x + 2y + z = 0, \quad x + 4y + z = 0, \quad 2x + y + 4z = 0 \] is given, then:

• A. {it is inconsistent}
• B. {it has only the trivial solution \( x = 0, y = 0, z = 0 \)}
• C. {it can be reduced to a single equation and so a solution does not exist}
• D. {the determinant of the matrix of coefficients is zero}

Hint:

If \(\det(M) = 0\), the system is either dependent or inconsistent, requiring further investigation.

Answer 1

The Correct Option is D

Step 1: Forming the coefficient matrix. \[ M = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 4 & 1 \\ 2 & 1 & 4 \end{bmatrix} \] Step 2: Computing determinant. \[ \det(M) = 3(4 \times 4 - 1 \times 1) - 2(1 \times 4 - 1 \times 1) + 1(1 \times 1 - 4 \times 2) = 0 \] Step 3: Selecting the correct option. Since the determinant is zero, the system is either inconsistent or has infinitely many solutions.