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A necessary and sufficient condition that values, not all zero may be assigned to n variables x1, x2, ......., xn so that the homogeneous equations, ai1x1 + ai2x2 + ..... + ainxn = 0 (i = 1, 2,.....n) hold simultaneously, is :
- a)|aij|n × n = 0
- b)|aij|n × n ≠ 0
- c)x1, x2, ....., xn is a linear independent set
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A necessary and sufficient condition that values, not all zero may be ...
For the scalars, ai1, ai2, ..........., ain (i,= 1, 2,.....,n)
not all zero such that ai1x1 + ai2x2 + ...... + ain xn = 0
⇔ that {x1, x2,..........,xn} is linearly dependent.
⇒ Rank of coefficient matrix will be less than. Hence, minor of order n in the matrix or determinant of the coefficient matrix will be zero.
not all zero such that ai1x1 + ai2x2 + ...... + ain xn = 0
⇔ that {x1, x2,..........,xn} is linearly dependent.
⇒ Rank of coefficient matrix will be less than. Hence, minor of order n in the matrix or determinant of the coefficient matrix will be zero.
The correct answer is: |aij|n × n = 0
Most Upvoted Answer
A necessary and sufficient condition that values, not all zero may be ...
The necessary and sufficient condition for the homogeneous equations to hold simultaneously is that the determinant of the matrix A, formed by the coefficients aij, is zero.
In other words, if det(A) = 0, then there exist non-zero values that can be assigned to the variables x1, x2, ..., xn such that the homogeneous equations hold.
Conversely, if det(A) ≠ 0, then for any assignment of values to the variables x1, x2, ..., xn, the homogeneous equations will not hold.
Therefore, the necessary and sufficient condition is:
det(A) = 0
In other words, if det(A) = 0, then there exist non-zero values that can be assigned to the variables x1, x2, ..., xn such that the homogeneous equations hold.
Conversely, if det(A) ≠ 0, then for any assignment of values to the variables x1, x2, ..., xn, the homogeneous equations will not hold.
Therefore, the necessary and sufficient condition is:
det(A) = 0
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A necessary and sufficient condition that values, not all zero may be assigned tonvariablesx1,x2, .......,xn so that the homogeneous equations, ai1x1 + ai2x2 + ..... + ainxn = 0(i= 1, 2,.....n) hold simultaneously, is :a)|aij|n × n= 0b)|aij|n × n≠ 0c)x1,x2, .....,xnis a linear independent setd)None of theseCorrect answer is option 'A'. Can you explain this answer?
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