Question

Linear algebra question see attachment please.

Answer 1

Problem 3.6

Answer 2

what do you know about the relation betwen the equation `Ax=0` and the invertibility of matrix, A ?

Answer 3

If Ax=0 then the matrix is non-invertible

Answer 4

Correct?

Answer 5

you must be knowing that the column vectors in matrix A are `linearly dependent` if `Ax=0` has a solution other than the zero vector, right ?

Answer 6

That part I am not familiar with

Answer 7

Could you explain?

Answer 8

you seem to be having the right idea, let me put your statement more precicely :

If Ax=0 has a nonzero solution, then the matrix is non-invertible

Answer 9

What I know is that if Ax=0 has a solution other than zero the solution is not unique, and it has free variables. I did not know that the column vectors were linearly dependent, nor did I know that if Ax=0 has a nonzero solution that the matrix was non invertible

Answer 10

I'll let your professor teach you about linearly `independent / dependent` stuff

For this problem. you just need to know this : `Ax = 0 has a nonzero solution only if the matrix A is not invertible`

Answer 11

Ok

Answer 12

A is not invertible, so the determinant of A must be 0

Answer 13

set the determinant equal to 0, you get a quadratic which you can solve for \(\lambda\)

Answer 14

Ok but can you explain why Ax=0 has a nonzero solution only if the matrix A is non invertible?

Answer 15

you need to know about the definitions of linearly independent and dependent vectors stuff, i prefer you not learning this from me...

Answer 16

http://mathworld.wolfram.com/LinearlyDependentVectors.html

Answer 17

By definition, if `Ax=0` has a nonzero solution, then the column vectors in A are said to be "linearly dependent"

Answer 18

Once you get the idea of "linearly dependence", you can try thinking of why a matrix with linearly dependent column vectors cannot be invertible