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

In summary, the determinant of a matrix [a1 b1; a2 b2] indicates the existence and uniqueness of a solution for a system of 2 equations in 2 unknowns. If the determinant is nonzero, the solution exists and is unique. However, if the determinant is zero, further checking the determinants of [a1 c1; a2 c2] and [b1 c1; b2 c2] is necessary. If both determinants are zero, the system has infinitely many solutions. If at least one of the two determinants is nonzero, the system has no solution. The bolded statement is correct and can be proven using linear algebra. This concept is also used in studying solvability
  • #1
kingwinner
1,270
0
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!
 
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  • #2
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"]
 

1. What is a system of linear equations?

A system of linear equations is a set of two or more equations that involve two or more variables. These equations can be solved simultaneously to find the values of the variables that satisfy all of the equations.

2. How can you tell if a system of linear equations has a unique solution?

A system of linear equations has a unique solution if the equations intersect at one point, meaning there is one set of values for the variables that satisfies all of the equations. This can be graphically represented by two lines intersecting at one point.

3. What does it mean if a system of linear equations has infinitely many solutions?

If a system of linear equations has infinitely many solutions, it means that the equations are essentially the same or are multiples of each other. This results in all points on the line being a solution to the system of equations.

4. How can you determine if a system of linear equations has no solutions?

A system of linear equations has no solutions if the equations are parallel and do not intersect. This means that there are no values for the variables that can satisfy all of the equations at the same time.

5. Is it possible for a system of linear equations to have more than one unique solution?

No, a system of linear equations can only have one unique solution. This is because each equation represents a specific line and the intersection of these lines can only occur at one point.

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