Examining Augmented Matrices: Understanding Solutions and Inconsistencies

[pyroknife]

I attached the problem.

I just wanted to check if I'm thinking about a few of these parts correctly.

a) yes because the coefficient matrix has 2 rows that are multiples of each other, thus det=0
b)Yes, but the system does not have a unique solution b/c 2 rows are multiples of each other in the coefficient matrix.
c) Imma skip this
d) skipping this
e) I think they worded this problem wrong. if det(coefficient matrix)=0, then there exists either inifinite many solutions OR NO solutions.
f) skip

 

[jbunniii]

I attached the problem.

I just wanted to check if I'm thinking about a few of these parts correctly.

a) yes because the coefficient matrix has 2 rows that are multiples of each other, thus det=0

Correct.

b)Yes, but the system does not have a unique solution b/c 2 rows are multiples of each other in the coefficient matrix.

Consider what happens if, say, a = 1 and d = 1.

e) I think they worded this problem wrong. if det(coefficient matrix)=0, then there exists either inifinite many solutions OR NO solutions.

And whether there are infinitely many solutions or no solutions depends on a, b, c, d. I agree that the wording of this question is strange.

 

[HallsofIvy]

It is unfortunate that you skipped over c and d because they are essential to e and f! d asked you to tell what must be true of the last column in order that the system be "inconsistent". e asks you to tell wheter or not, in that case, there are an infinite number of solutions or no solution. What does "inconsistent" mean?

f happens to give you a set of values in which the last number is twice the first number. Do you see why that is important?

 

[pyroknife]

Consider what happens if, say, a = 1 and d = 1.

Well since 2 rows in the coefficient matrix are multiples of each other, the system is either going to have a free variable, thus one option is infinite solutions. But if it's inconsistent, then there's no solutions. I don't think it's possible for the system to have a unique solution independent of a, b, c, d.

It is unfortunate that you skipped over c and d because they are essential to e and f! d asked you to tell what must be true of the last column in order that the system be "inconsistent". e asks you to tell wheter or not, in that case, there are an infinite number of solutions or no solution. What does "inconsistent" mean?

f happens to give you a set of values in which the last number is twice the first number. Do you see why that is important?

Well I did them, posting my work would have just taken a while.