- #1
kingwinner
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Suppose we have a system of 2 equations in 2 unknowns x,y:
a1x+b1y=c1
a2x+b2y=c2
If the determinant of
[a1 b1
a2 b2]
is nonzero, then the solution to the system exists and is unique. [I am OK with this]
If the determinant of
[a1 b1
a2 b2]
is zero,
this does not distinguish between the cases of no solution and infinitely many solutions.
To gain some insight, we need to further check the determinatnts of
[a1 c1
a2 c2]
and
[b1 c1
b2 c2]
If both determinants are zero, then the system has infinitely many soltuions.
If at least one of the two determinants are nonzero, then the system has no solution.
===================================
Is the bolded part correct or not? I don't see why it is true. How can we prove it?
Thanks for any help!
a1x+b1y=c1
a2x+b2y=c2
If the determinant of
[a1 b1
a2 b2]
is nonzero, then the solution to the system exists and is unique. [I am OK with this]
If the determinant of
[a1 b1
a2 b2]
is zero,
this does not distinguish between the cases of no solution and infinitely many solutions.
To gain some insight, we need to further check the determinatnts of
[a1 c1
a2 c2]
and
[b1 c1
b2 c2]
If both determinants are zero, then the system has infinitely many soltuions.
If at least one of the two determinants are nonzero, then the system has no solution.
===================================
Is the bolded part correct or not? I don't see why it is true. How can we prove it?
Thanks for any help!
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