Linear Algebra help plox? (1 Viewer)

[hayabusaboston]

I just registered for a maths diploma and they accepted me only now after two full weeks so im trying to catch up on content

How u do dis?

2. Determine the values (if any) of t in R for which the following linear system has:
(a) no solutions,
(b) infinitely many solutions,
(c) a unique solution.
In the case (b) of infinitely many solutions, find all solutions.

tx-5y+3z=-8t
-tx+8y+3tz=11t
-x+2y+3z=1

 

[He-Mann]

Gaussian elimination.

 

[hayabusaboston]

Gaussian elimination.

can u help ploxx?

Im confused how to get to RRE with t's and numbers, + the rules for one, infinite and no solutions. would be helpful if someone did the question step by step so I understand for subsequent questions.

 

[He-Mann]

Do row operations as usual, group up the rows containing t to make it easier to analyse.:



There's probably a faster sequence of row operations to reach RREF, but that's one way.

Start with (a), what values of t produces a contradiction? That's when you cannot solve the system.

 

[hayabusaboston]

Sorry, I made an error. brb

Ok

 

[He-Mann]

Suppose you had a row like this: (0 0 0 | 2317). Would it make sense?

 

[hayabusaboston]

Do row operations as usual, group up the rows containing t to make it easier to analyse.:



There's probably a faster sequence of row operations to reach RREF, but that's one way.

Start with (a), what values of t produces a contradiction? That's when you cannot solve the system.

t=4 is that only solution that satisfies a)?

And t not equal to 4 for c)?

what about b)?

 

[hayabusaboston]

Suppose you had a row like this: (0 0 0 | 2317). Would it make sense?

Yea I know the no solution part i just was wondering about how to distinguish 1 vs infinite solutions. Something about not having an identity matrix isnt it? then infinite solutions? idk

 

[He-Mann]

Yea I know the no solution part i just was wondering about how to distinguish 1 vs infinite solutions. Something about not having an identity matrix isnt it? then infinite solutions? idk

If you want to get infinite solutions, then you need to prevent at least one row from giving any information. Intuition: for RREF, each row should give information about one of x, y, or z. But when you stop one row from finding either x, y, or z, then you won't be able to solve other rows. e.g., solving for y and z in y+z = 2 when you don't have any information on y or z. Thus, you get infinite solutions for y and z.

i.e., produce a zero row (0 0 0 | 0).

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Your solutions are correct for (a) and (c).

 

[He-Mann]

It seems that there is no t such that the system yields infinite solutions.