Mathematics Exam > Mathematics Questions > If 3x + 2y + z = 0, x + 4y+z = 0, 2x+ y+ 4z =...
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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 trivial solution
- c)it can be reduced to a single equation and so a solution does not exist
- d)the determinant of the matrix of coefficient 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 equatio...
Given:
The system of equations is:
1) 3x + 2y + z = 0
2) x + 4y + z = 0
3) 2x + y + 4z = 0
To find:
The nature of the system of equations.
Solution:
To determine the nature of the system of equations, we need to consider the coefficient matrix and solve it using the determinant method.
Determinant Method:
The system of equations can be written in matrix form as AX = 0, where A is the coefficient matrix and X is the column matrix of variables (x, y, z).
The coefficient matrix A is given by:
A = |3 2 1|
|1 4 1|
|2 1 4|
The determinant of the coefficient matrix is calculated as follows:
|A| = 3(4*4 - 1*1) - 2(1*4 - 2*1) + 1(1*1 - 2*4)
= 3(16 - 1) - 2(4 - 2) + 1(1 - 8)
= 3(15) - 2(2) + 1(-7)
= 45 - 4 - 7
= 34
Analysis:
If the determinant of the coefficient matrix (|A|) is zero, then the system of equations has either no solution or infinitely many solutions. If |A| is non-zero, then the system has a unique solution.
Conclusion:
Since the determinant of the coefficient matrix is non-zero (34 ≠ 0), the system of equations has a unique solution. Therefore, the correct answer is option 'B' - the system of equations has only a trivial solution.
The system of equations is:
1) 3x + 2y + z = 0
2) x + 4y + z = 0
3) 2x + y + 4z = 0
To find:
The nature of the system of equations.
Solution:
To determine the nature of the system of equations, we need to consider the coefficient matrix and solve it using the determinant method.
Determinant Method:
The system of equations can be written in matrix form as AX = 0, where A is the coefficient matrix and X is the column matrix of variables (x, y, z).
The coefficient matrix A is given by:
A = |3 2 1|
|1 4 1|
|2 1 4|
The determinant of the coefficient matrix is calculated as follows:
|A| = 3(4*4 - 1*1) - 2(1*4 - 2*1) + 1(1*1 - 2*4)
= 3(16 - 1) - 2(4 - 2) + 1(1 - 8)
= 3(15) - 2(2) + 1(-7)
= 45 - 4 - 7
= 34
Analysis:
If the determinant of the coefficient matrix (|A|) is zero, then the system of equations has either no solution or infinitely many solutions. If |A| is non-zero, then the system has a unique solution.
Conclusion:
Since the determinant of the coefficient matrix is non-zero (34 ≠ 0), the system of equations has a unique solution. Therefore, the correct answer is option 'B' - the system of equations has only a trivial solution.
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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 trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient 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 trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics 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 trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 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 trivial solutionc)it can be reduced to a single equation and so a solution does not existd)the determinant of the matrix of coefficient is zeroCorrect answer is option 'B'. Can you explain this answer?.
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