System of linear equations-unique, infinitely many, or no solutions


by kingwinner
Tags: equationsunique, infinitely, linear, solutions
kingwinner
kingwinner is offline
#1
Sep18-09, 11:30 AM
P: 1,270
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!
Phys.Org News Partner Science news on Phys.org
SensaBubble: It's a bubble, but not as we know it (w/ video)
The hemihelix: Scientists discover a new shape using rubber bands (w/ video)
Microbes provide insights into evolution of human language
kingwinner
kingwinner is offline
#2
Sep20-09, 01:46 AM
P: 1,270
Can someone "confirm or disprove" the bolded part, please?

[my PDE book seems to be applying these ideas from linear algebra to study the solvability of initial value problems for quasilinear partial differential equations, but I can't find those results in my linear algebra textbooks other than the result "det(A) is not 0 <=> solution exists and is unique"]


Register to reply

Related Discussions
Existential Proof of a Unique Solution to a Set of Non-Linear Equations General Math 3
Solution to system of linear equations in range of system matrix Calculus & Beyond Homework 2
Linear Algebra (finding values of a constant k to give no/unique/infinite solutions) Calculus & Beyond Homework 6
solutions of system of nonlinear equations Linear & Abstract Algebra 14
Solutions to a system of Linear Equations Calculus & Beyond Homework 4