Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Questions > Given system of linear equations3x + 2y + z =...
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Given system of linear equations
3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0, has
- a)no solution
- b)only trivial solution
- c)Infinite solution
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Given system of linear equations3x + 2y + z = 0, x + 4y + z = 0, 2x + ...
To determine the nature of the solution for the given system of linear equations, we can use the concept of matrices and determinants. Let's solve the system of equations using matrix notation.
The given system of linear equations can be written in matrix form as:
```
[3 2 1] [x] [0]
[1 4 1] [y] = [0]
[2 1 4] [z] [0]
```
We can represent this system as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Now, to determine the nature of the solution, we need to find the determinant of the coefficient matrix A.
Determinant of A:
|A| = |3 2 1|
|1 4 1|
|2 1 4|
Using the expansion method, we can evaluate the determinant as follows:
|A| = (3 * (4 * 4 - 1 * 1)) - (2 * (1 * 4 - 1 * 2)) + (1 * (1 * 2 - 4 * 1))
= (3 * 15) - (2 * 2) + (1 * (-2))
= 45 - 4 - 2
= 39
Since the determinant of A is non-zero (39 ≠ 0), we can conclude that the system of equations has a unique solution.
Therefore, the correct answer is option C: Infinite solution.
The given system of linear equations can be written in matrix form as:
```
[3 2 1] [x] [0]
[1 4 1] [y] = [0]
[2 1 4] [z] [0]
```
We can represent this system as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Now, to determine the nature of the solution, we need to find the determinant of the coefficient matrix A.
Determinant of A:
|A| = |3 2 1|
|1 4 1|
|2 1 4|
Using the expansion method, we can evaluate the determinant as follows:
|A| = (3 * (4 * 4 - 1 * 1)) - (2 * (1 * 4 - 1 * 2)) + (1 * (1 * 2 - 4 * 1))
= (3 * 15) - (2 * 2) + (1 * (-2))
= 45 - 4 - 2
= 39
Since the determinant of A is non-zero (39 ≠ 0), we can conclude that the system of equations has a unique solution.
Therefore, the correct answer is option C: Infinite solution.
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Given system of linear equations3x + 2y + z = 0, x + 4y + z = 0, 2x + ...
B
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Given system of linear equations3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0, hasa)no solutionb)only trivial solutionc)Infinite solutiond)None of theseCorrect answer is option 'C'. Can you explain this answer?
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