Question

Use matrix Method to prove the following lines are concurrent
2x+5y+19 = 0
4x-3y-1=0
5x-2y+4=0

Answer 1

find the determinant. It should be =0 if this lines are concurrent

Answer 2

? can you be a little more specific please ?

Answer 3

Determinant of the coefficient matrix.

Answer 4

... little bit more please.

Answer 5

If a matrix is singular it is not invertible.

Answer 6

? little bit more please :)

Answer 7

It is helpful if you let us know what your thoughts on it are so far; this will prevent people from telling you things you already know, and help them write their answers at an appropriate level-Thanks.

Answer 8

im not to sure. but is the determinant like the 2 and 5 ?

Answer 9

you look preaty raw on this topic, just read something about matrix algebra and it's aplication to linear system of equations

Answer 10

I am not sure either of what you are trying to say, there is only one possible determinant.

Answer 11

Like FFM said, write down the coefficients of x, y and z in separate columns of a determinant(3 rows and 3 columns). Then find the determinant's value(by expansion). If it comes out to be non-zero, the lines are concurrent.

Answer 12

So the co-efficents are 2, 5, 19
4,3, 1
5, 2 ,4 ?

Answer 13

2,5,19
4,-3,-1
5,-2,4
This would be the determinant. Can you evaluate it by expansion?

Answer 14

"If it comes out to be *zero*, the lines are concurrent."

Answer 15

But if determinant is zero,then the matrix is not invertible, so it wouldn't be possible to solve it!

Answer 16

LOL, so if its zero, they are all concurrent, thats it ?

Answer 17

We don't need to solve it. If it's non-zero then it will give the area of the corresponding parallelogram.

Answer 18

Here, the rank is 2, this matrix is going to have non-trivial solutions.

Answer 19

think about it this way. Since this determinant is giving the are of a triangle formed by this lines, if the lines are concurrent they form no triangle, so area is 0

Answer 20

I would row reduce the matrix.

Answer 21

Oh you're right. Thanks FFM and Myko!

Answer 22

a zero determinant does not imply concurrent

Answer 23

it is possible that there is no solution (though this one has a solution)

no solution could give a zero determinant

Answer 24

@Zarkon, In this case the determinant does come out to be zero. I was thinking that if the lines are concurrent, it means that they have a unique solution. So determinant would be non-zero.

Answer 25

there is a unique solution...you have an augmented 3x3...not a system of 3 equations and 3 unknowns

Answer 26

the determinant should be be used in this type of problem...one should use elementary row operations to get the matrix in rref

Answer 27

*should NOT be used

Answer 28

sry, but i desagree. Determinant makes it much easyer and faster. :) YTou can find a proof of this anywhere. Look here for example:
http://www.math.wisc.edu/~robbin/461dir/coordinateGeometry.pdf

Answer 29

@Zarkon

Answer 30

@Mani_Jha

Answer 31

so. It would be
2 5 19
2 -3 -1
5 -2 4
then what do i multiply it with ? im still confused.

Answer 32

just find the determinant. If it's 0 lines are concurrent, if not--- not

Answer 33

i got a determinant of -506 ? did i do something wrong lol ?

Answer 34

?

Answer 35

SEC I CHECK

Answer 36

you are wrong it's equal 0
|2 5 19|
|4 -3 -1| = -24-152 -25 +285 -80 -4 = 0
|5 -2 4|

Answer 37

@axle1

Answer 38

awesome, thanks for the help man! really appreciate it :)

Answer 39

The OP probably meant collinear. This system clearly has non-trivial solutions.

Answer 40

@myko
suppose that you have 3 equation similar to the above...

it is possible that once you put it in a matrix like you did here you could end up with

\[\left[\begin{matrix}1 & 1 & 0\\ 0 &0 & 1\\0 & 0 & 0\end{matrix}\right]\]

for example the system below

x+y=1
2x+2y=3
x-y=1

this system has no solution and thus is not

concurrent

but its determinant is zero

Answer 41

I ment the system

x+y=1
2x+2y=3
x+y=2

Answer 42

it's a special case, becouse your first 2 columns are dependent, so span same line

Answer 43

But the concurrent means unique solution right?

Answer 44

right

Answer 45

yes..unique solution

Answer 46

the above does not have a unique solution...thus a determinant of zero does not tell me that are concurrent

Answer 47

Here we don't have non-trivial solutions so this can't be concurrent ..

Answer 48

I mean we have.

Answer 49

and when I say the above...I mean the equatiuon i made up

Answer 50

I was referring to the OP's system.

Answer 51

the original equation of the op has a unique solution...but not because the determinant is zero

Answer 52

x+y=1
2x+2y=3
x+y=2

you don't get that from the matrix you wrote....

Answer 53

rew reduce

\[\left[\begin{matrix}1 & 1 & 1\\ 2 &2 & 3\\1 & 1 & 2\end{matrix}\right]\]

Answer 54

that is the augmented matrix

Answer 55

I don't understand, how come the system has unique solution?

This is homogeneous system, for unique solution it implies the trivial solution. But the rank of the coefficient matrix is 2 implying non-trivial solutions, ergo making non-unique solution.

where I am going wrong?

Answer 56

i need to learn to spell...*row reduce

Answer 57

row reduce the original matrix...you will get a unique solution

Answer 58

x=-2 and y=-3

Answer 59

Zarkon, for a homogeneous system, if the rank of the augmented matrix is less than the number of unknowns that implies non-trivial solution.

By Rouché–Capelli theorem.

Answer 60

http://www.wolframalpha.com/input/?i=2x%2B5y%2B19+%3D+0%2C+4x-3y-1%3D0%2C+5x-2y%2B4%3D0

Answer 61

you don't have a homogenous system

Answer 62

Crap! Sorry I thought this system has 3 unknowns :(

Answer 63

are we done now :)