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If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, 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)determinant of the matrix of coefficients is zero
Correct answer is option 'B'. Can you explain this answer?
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If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equ...
Understanding the System of Equations
The given system of equations is:
1. 3x + 2y + z = 0
2. x + 4y + z = 0
3. 2x + y + 4z = 0
To analyze this system, we can represent it in matrix form and determine its solution.
Matrix Representation
The equations can be represented in matrix form as:
\[
\begin{bmatrix}
3 & 2 & 1 \\
1 & 4 & 1 \\
2 & 1 & 4
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
\]
Calculating the Determinant
To find if the system has only the trivial solution, we calculate the determinant of the coefficient matrix:
\[
\text{det} = 3(4 \cdot 4 - 1 \cdot 1) - 2(1 \cdot 4 - 1 \cdot 2) + 1(1 \cdot 1 - 4 \cdot 2)
\]
Calculating this, we find that the determinant is non-zero, indicating that the system has only the trivial solution.
Conclusion
Since the determinant is non-zero, the only solution to the equations is:
- x = 0
- y = 0
- z = 0
Thus, the correct answer is option 'B': it has only the trivial solution x = 0, y = 0, z = 0.
The given system of equations is:
1. 3x + 2y + z = 0
2. x + 4y + z = 0
3. 2x + y + 4z = 0
To analyze this system, we can represent it in matrix form and determine its solution.
Matrix Representation
The equations can be represented in matrix form as:
\[
\begin{bmatrix}
3 & 2 & 1 \\
1 & 4 & 1 \\
2 & 1 & 4
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
\]
Calculating the Determinant
To find if the system has only the trivial solution, we calculate the determinant of the coefficient matrix:
\[
\text{det} = 3(4 \cdot 4 - 1 \cdot 1) - 2(1 \cdot 4 - 1 \cdot 2) + 1(1 \cdot 1 - 4 \cdot 2)
\]
Calculating this, we find that the determinant is non-zero, indicating that the system has only the trivial solution.
Conclusion
Since the determinant is non-zero, the only solution to the equations is:
- x = 0
- y = 0
- z = 0
Thus, the correct answer is option 'B': it has only the trivial solution x = 0, y = 0, z = 0.
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If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, thena)it is inconsistentb)it has only the trivial solution x = 0, y = 0, z = 0c)it can be reduced to a single equation and so a solution does not existd)determinant of the matrix of coefficients is zeroCorrect answer is option 'B'. Can you explain this answer?
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If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, thena)it is inconsistentb)it has only the trivial solution x = 0, y = 0, z = 0c)it can be reduced to a single equation and so a solution does not existd)determinant of the matrix of coefficients is zeroCorrect answer is option 'B'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, thena)it is inconsistentb)it has only the trivial solution x = 0, y = 0, z = 0c)it can be reduced to a single equation and so a solution does not existd)determinant of the matrix of coefficients is zeroCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, thena)it is inconsistentb)it has only the trivial solution x = 0, y = 0, z = 0c)it can be reduced to a single equation and so a solution does not existd)determinant of the matrix of coefficients is zeroCorrect answer is option 'B'. Can you explain this answer?.
If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, thena)it is inconsistentb)it has only the trivial solution x = 0, y = 0, z = 0c)it can be reduced to a single equation and so a solution does not existd)determinant of the matrix of coefficients is zeroCorrect answer is option 'B'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, thena)it is inconsistentb)it has only the trivial solution x = 0, y = 0, z = 0c)it can be reduced to a single equation and so a solution does not existd)determinant of the matrix of coefficients is zeroCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, thena)it is inconsistentb)it has only the trivial solution x = 0, y = 0, z = 0c)it can be reduced to a single equation and so a solution does not existd)determinant of the matrix of coefficients is zeroCorrect answer is option 'B'. Can you explain this answer?.
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