System of linear equationsunique, infinitely many, or no solutionsby kingwinner Tags: equationsunique, infinitely, linear, solutions 

#1
Sep1809, 11:30 AM

P: 1,270

Suppose we have a system of 2 equations in 2 unknowns x,y:
a_{1}x+b_{1}y=c_{1} a_{2}x+b_{2}y=c_{2} 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! 



#2
Sep2009, 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"] 


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