MATH1131 Linear Algebra: Chapter 4 Problem 17

In this problem we determine values of unknown constant k, if any, will give unique solution, no solution infinitely many solutions to the system of equations. Presented by N J Wildberger of the School of Mathematics and Statistics, Faculty of Science, UNSW.

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[@Xsjr03]

I don't get it, our teacher told us that the system is solvable if the rank of the coefficient matrix equals the rank of the extended matrix. In this case, wether you put k = 3 or k # 3 doesn't make a difference since in both instances, the ranks of the two matrices are equal, therefore you should have a solvable system in either case. And then she told us if the number of variables (x,y,z here) is equal to the rank of the coefficient matrix, then you have a unique solution. If the variables are more than the rank, then you have infinite solutions. So here, if k = 3 it should mean we have infinite solutions, and when k # 3 then we have a unique solution. Where am i going wrong?

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[@eliekhoury9819]

what if you took if k=0 wouldn't that give you -3=3? so we will get the no solution option

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[@jeromeblacq7528]

what about k = -3

[@mery.21st]

Unique soln, since it becomes -6z=6

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[@MultiDylhan]

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[@maahasgharali571]

How do you know till how long you have to keep on reducing the matrix. Like when u said at 4.49 "we are in a position to figure out..." how do you know. thanks in advance :)

[@lucaspaulson8132]

gah... I've been trying to figure that out all afternoon.

[@purplerain5305]

He reduces it to row echelon form, so then he is is a position to use the last row to find out those values for k.

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[@koenalamola6020]

what if k=0 for no solution?

[@jasonchanjinan]

if k = 0 then you will have a unique solution. For the system to be inconsistent, the L.H.S must be zero and then R.H.S must be non zero. However, in this case for k-3 to be zero, k must be equal to 3, but this also makes 3-k equal to 0.

[@ozz961]

it's impossible no solution case , in order to the solution to be unique it must be equal to a unique value ex: A.X3 = b , no solution ex: 0 = b

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[@mery.21st]

Thank you.