Question:
If the system of equations:
\[
3x + 2y + z = 0, \quad x + 4y + z = 0, \quad 2x + y + 4z = 0
\]
is given, then:
If the system of equations:
\[
3x + 2y + z = 0, \quad x + 4y + z = 0, \quad 2x + y + 4z = 0
\]
is given, then:
Show Hint
If \(\det(M) = 0\), the system is either dependent or inconsistent, requiring further investigation.
Updated On: Feb 4, 2025
- {it is inconsistent}
- {it has only the trivial solution \( x = 0, y = 0, z = 0 \)}
- {it can be reduced to a single equation and so a solution does not exist}
- {the determinant of the matrix of coefficients is zero}
Show Solution
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The Correct Option is D
Solution and Explanation
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.
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