Question

for the matrix below
The system has more than one solution when k =
The system has no solution when k =

Answer 1

hey @ganeshie8 could you please help me with this question aswell?

Answer 2

its ohk i hve figured out the answer

Answer 3

well , no SOLUTIONS means lines are skew
more than one solution (infinitly many somlution case ) means lines are parallel

Answer 4

:3 cool

Answer 5

oh wait sorry! can u please explain the matrix one, i thought i wrote another question on here

Answer 6

that works too ^^
start by rowreducing the matrix

Answer 7

\[
\left[
\begin{array}{cccc}
1&1&k& | &8\\
1&k&1&|&5 \\
k&1&1&|&3

\end{array}
\right]
\]

Answer 8

nothing came up? ^^^ just letters n numbers

Answer 9

after row reducing, mess with the last row to find the values of k that make the system having infinitely/no solutions

Answer 10

what do you get after rowreducing ?

Answer 11

how would i row reduce this one? with k included in it

Answer 12

as usual, just think of k as some value :

\[
\left[
\begin{array}{cccc}
1&1&k& | &8\\
1&k&1&|&5 \\
k&1&1&|&3

\end{array}
\right] \\~\\

R2-R1\left[
\begin{array}{cccc}
1&1&k& | &8\\
0&k-1&1-k&|&-3 \\
k&1&1&|&3

\end{array}
\right] \\

\]

Answer 13

keep going

Answer 14

\[
\left[
\begin{array}{cccc}
1&1&k& | &8\\
1&k&1&|&5 \\
k&1&1&|&3

\end{array}
\right] \\~\\

R2-R1\left[
\begin{array}{cccc}
1&1&k& | &8\\
0&k-1&1-k&|&-3 \\
k&1&1&|&3

\end{array}
\right] \\~\\

R3-kR1\left[
\begin{array}{cccc}
1&1&k& | &8\\
0&k-1&1-k&|&-3 \\
0&1-k&1-k^2&|&3-8k

\end{array}
\right] \\

\]

Answer 15

this is what came up

Answer 16

there seeems to be some problem with latex, il take a screenshot and attach wait

Answer 17

thanks

Answer 18

http://prntscr.com/4bdanc

Answer 19

ohk

Answer 20

i get what you mean

Answer 21

\[
\left[
\begin{array}{cccc}
1&1&k& | &8\\
1&k&1&|&5 \\
k&1&1&|&3

\end{array}
\right] \\~\\

R2-R1\left[
\begin{array}{cccc}
1&1&k& | &8\\
0&k-1&1-k&|&-3 \\
k&1&1&|&3

\end{array}
\right] \\~\\

R3-kR1\left[
\begin{array}{cccc}
1&1&k& | &8\\
0&k-1&1-k&|&-3 \\
0&1-k&1-k^2&|&3-8k

\end{array}
\right] \\~\\
R3+R2\left[
\begin{array}{cccc}
1&1&k& | &8\\
0&k-1&1-k&|&-3 \\
0&0&2-k^2-k&|&-8k

\end{array}
\right] \\~\\

\]

Answer 22

http://prntscr.com/4bdbel

Answer 23

is it row reduced yet?

Answer 24

since it is upper triangular, we're done with row reducing.

Answer 25

gani u mean, det =0
diagonal multi is det ?

Answer 26

next, we need to interpret the last row

Answer 27

what do you mean since its upper triangle?

Answer 28

the goal of row-reduction is to get an upper triangular matrix

Answer 29

http://1.bp.blogspot.com/_ue1VUbFmrsE/TEYpgdt_A8I/AAAAAAAAAEQ/JbyOjk7JN7U/s1600/Matrix_5.bmp

Answer 30

upper triangular matrix has all 0's below the diagonal ^^

Answer 31

@ikram002p im thinking of finding the k values that make the last row all 0's etc...

Answer 32

ohkk

Answer 33

all 0's gives u no solution
but det=0 gives u the trivial solution
i think se asked for both cases

Answer 34

now i have to make 2-k^2 -k+8k=0?

Answer 35

For no solutions, just solve 2-k^2-k = 0

Answer 36

k=1or -2

Answer 37

yep! those are the only values of k that produce no solutions to the system

Answer 38

now for trivial :o

Answer 39

ohk and when infinte ?

Answer 40

2-k^2 -k =8k <--have to solve this for k?

Answer 41

not exactly... that gives an unique solution, so no.

for more than 1 solution(infinitely many) , ALL the elements on last row must equal 0 :

2-k^2-k = 0
-8k=0

Answer 42

notice that you cannot get both the elements 0 at the same time,
so we cannot have more than 1 solution for this system no matter what the value of k is

Answer 43

ohk right i understand, thanks again for all the help! u saved me so much time!

Answer 44

np :) this is a very special matrix ! notice that it is symmetric and it has constant value in the other diagonal... these matrices are the most important matrices in engineering.

Answer 45

really ?!

Answer 46

what application could be ?

Answer 47

got that !

Answer 48

is there another way you can solve this problem? cause when i was searching it before, some examples were using determinants

Answer 49

watch this video @ikram002p for more info
https://www.youtube.com/watch?v=CgfkEUOFAj0

Answer 50

ima delete my earlier comment as it is not correct 100%

Answer 51

i watched it before.

Answer 52

okie, lets look at another method using determinant

Answer 53

what engineering course is that for? 1st year 2nd year?

Answer 54

personally i prefer rowreducing to determinants cuz rowreducing is simple+quickest

Answer 55

yes this was does seem more simple to understand, the other looked complicated and long

Answer 56

Linear Algebra and Differential equations are prerequisite for that course, so i would guess it is not an undergraduate level course

Answer 57

what course ?

Answer 58

more info about the course here :
http://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008/

Answer 59

OMG !
where uve been when i was searching for such a course xD

Answer 60

lol im thinking of doing this course after finishing problemsets in LA

Answer 61

im actually doing my first year in engineering and didnt take advanced calculus in highschool

Answer 62

ull gonna take them in ur first yr ,
ganesh hehe was looking for application courses xD
and i already taken pre courses
i searched in physics department , but found tough stuff
more of physics concepts