Augmented matrix

[solemn sparrow]

i couldn't solve this beyond (b)

[median horizon]

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[solemn sparrow]

this is my work

i couldn't solve this any further

[rain glade]

Hint: look at the last row

[solemn sparrow]

yeah ?

w8

give me a sec

let me think

[rain glade]

Notice how in this form:
$\begin{pmatrix}
1 & 0 & -5 \
0 & \lambda & -3\
0 & 0 & a
\end{pmatrix}\cdot
\begin{pmatrix}
x\
y\
z
\end{pmatrix} = \begin{pmatrix}
C_1\
C_2\
C_3
\end{pmatrix}$

[quartz palm]

Lumberdude #LeaveWolfAsHeIs

[solemn sparrow]

whats a ?

[rain glade]

It's the entry you had there

[solemn sparrow]

okay

[rain glade]

What does the matrix having just one entry on the bottom row mean?

[solemn sparrow]

(z)(a) = C

[rain glade]

Exactly

So when would that not give an actual solution?

[solemn sparrow]

(z)(a) ! = C

[rain glade]

When is that the case?

[solemn sparrow]

evey other case than (z)(a) = C

[rain glade]

Hmmmm

Z is the variable in this case right?

[solemn sparrow]

yes

a is the constant

[rain glade]

So z =c/a is a solution

What happens if either of these constants is zero?

[solemn sparrow]

undefined

[rain glade]

So that means that?

[solemn sparrow]

there is no solution

[rain glade]

Perfect!

Which one has to be zero for that to be the case?

And what happens if both are zero?

[solemn sparrow]

the denominator

[solemn sparrow]

it's undefined

[rain glade]

Are you sure? 0*z=0

For what z is that true?

[solemn sparrow]

all

[rain glade]

Exactly, so infinitely many z satisfy this condition

[solemn sparrow]

okay i did the algebra

i can't find a real lambda

i can't factor it

[rain glade]

Give me one second

I will try to see if your work is correct so far

Your top row is incorrect

[solemn sparrow]

really

[rain glade]

So the rest will probably be incorrect too

[solemn sparrow]

okay

[rain glade]

Yeah it should equal -4 not -6

[solemn sparrow]

oh

you are right

😭

i didn't notice that

[rain glade]

That's ok, you noticed that Vinland saga is goated at least

These kind of mistakes will always happen!

[solemn sparrow]

yeah i think i can do it now

[rain glade]

I got the solution for "no real solution" and for "infinitely many"

[solemn sparrow]

this is what happens when you do math while sleep deprived

[rain glade]

Also for one solution

[rain glade]

Yeah take care of yourself man

Sleep deprivation can do you in

[solemn sparrow]

found the values for no solutions

-2 and -3

[rain glade]

Are you sure those are both yielding no solutions?

Hint, check the last row

[solemn sparrow]

let me post my new augmented matrix

actually -2 gives us infinite solutions

[rain glade]

Yes!

And -3?

[solemn sparrow]

-3 doesn't

0 != -3

[rain glade]

Good, I agree

Ok so we determined for what lambda we get no solution and for what lambda we get infinite solutions.

When is there exactly one solution?

[solemn sparrow]

lambda != 0 and lambda = C

3

3 gives a unique solution

[rain glade]

Are there more numbers for which this true?

[solemn sparrow]

i guess -2 ?

[rain glade]

Didn't you determine that -2 yielded infinitely many solutions?

[solemn sparrow]

yeah

[rain glade]

So that can't be one that yields only one solution, right?

[solemn sparrow]

other than 3 idk

excuse my shitty handwriting

[rain glade]

Huh, didn't you say -3 and -2?

[solemn sparrow]

-3 yields no solutions

[rain glade]

What is this exactly?

[solemn sparrow]

i set the bottom last row (z coeffecient) to equal C

[rain glade]

And C was?

Is that the bottom entry in the constant vector?

[solemn sparrow]

fourth column bottom row

[rain glade]

So you're saying c*z=c gives not exactly one solution?

[solemn sparrow]

i guess

i think i am as confused

as you are

[rain glade]

Ok, we determined what values give infinite and no solutions, those won't give unique solutions

Is there other values we can exclude?

Think of things like the determinant being 0

Because that is the only other thing that could bring us no solution, right?

[solemn sparrow]

okay

[rain glade]

Otherwise there should always be singular solutions

[solemn sparrow]

yeah

i think the question asks for one example for each

actually

no

it's pretty vague

[solemn sparrow]

-1

[rain glade]

I think it wants you to tell when each of these cases occur. If there are infinitely many $\lambda \in \real, \lambda \neq -2,-3$

[quartz palm]

Lumberdude #LeaveWolfAsHeIs

[rain glade]

They'd want you to mention that

[solemn sparrow]

yes

[rain glade]

Are you sure? Don't forget that you multiply the last row by the vector to yield the bottom constant

We have already excluded all values for which this system will not have a unique solution, bare the case when det(A)=0

[solemn sparrow]

you are right

-1 gives a unique solution actually

[rain glade]

Yes, any matrix that is non-singular, does not have a zero row equals some constant and where a row does not give 0*x=0 has a unique solution. I believe

[solemn sparrow]

i think that's it

inf solution = -3
no solution = -2
unique solution = any other value

i am kinda sure

[rain glade]

Yes you are correct!!

Well done

[solemn sparrow]

thank you man

thank you so much

[rain glade]

No problem!

Sorry for being confusing at times lmao

[solemn sparrow]

this is what i get for doing comp

[rain glade]

Computer science?

[solemn sparrow]

yes

[rain glade]

Cool!

Computer science seems kinda fun I'm ngl

[solemn sparrow]

you a math major ?

[rain glade]

No, I'm a chemical engineering student in grad school

[solemn sparrow]

oh

[solemn sparrow]

https://tenor.com/view/waltwhite-breakingbad-say-my-name-gif-7259290

[rain glade]

I did notice something that could have saved a lot of headache, the determinant of the matrix yields the solution too, due to the fact that it determines when a matrix is singular if it is 0...

[solemn sparrow]

oh yeah

[solemn sparrow]

+close