Question

Given A =
[ 3 -2
-6 4
12 -8 ], find one nontrivial solution Ax = 0 by inspection.

Answer 1

More specifically, what is meant by inspection?

Answer 2

by inspection just means just by looking at it

Answer 3

I'm not sure what exactly to look for, but I know that there's going to be infinitely many solutions (there's more rows than columns, so there's gonna be a row without pivots).

Answer 4

...wait, does that mean that since there's not a pivot in every column, that I could just throw any solution in and it would work?

Answer 5

every row*

Answer 6

i think the matrix you have shown is incorrect

have you considered x=[0; 0] yet?

Answer 7

But I'm looking for a nontrivial solution.

Answer 8

are you sure you're not missing a negative sign on the top row?

Answer 9

Whoops! You're right.

Answer 10

...waitaminute.

Answer 11

The second two rows can be reduced to zero rows.

Answer 12

if you look at the first row, then look at the last 2 rows
then realize that the last 2 rows are multiples of the 1st row

then solve for just the first row

Answer 13

So now we've got [ 3 -2; 0 0; 0 0;].

Answer 14

But I feel like I'm still forgetting a step.

Answer 15

I don't think x = [3 -2] is the solution.

Answer 16

x=[x;y]

now solve for x and y for
Ax=0

Answer 17

\[x=\left(\begin{matrix}x_1 \\ x_2\end{matrix}\right)\]

\[(3 -2)\left(\begin{matrix}x_1 \\ x_2\end{matrix}\right)=0\]

Answer 18

So x1 and x2 would have to be zero, then?

Answer 19

you said nontrivial

Answer 20

basically make up a number for x1, then solve for x2

then rewrite x

Answer 21

Ok, give me a second here...

Answer 22

So I could set x1 = 2 and x2 = 3, which would mean the result would be (3*2)+(3*-2)=6-6=0, which means the solution would be x = [2 3]?

Answer 23

and also [4 6] and [8 12] etc etc etc

Answer 24

awesome, thanks for the help!