Question

determine if the matrix is invertible

3 0 0
-3 -4 0
8 5 -3

Can we refer to the Invertible matrix theorem:
well is it safe to say that the neither columns of the matrix are multiples of the other columns , then it is linearly independent .????

Answer 1

if the derivative is non-zero the matrix is invertible

Answer 2

yes
- you mean the determinant

Answer 3

of course :P

Answer 4

but what about linear ind. ?

Answer 5

Yes, they are all the same idea, hold on I have a good list of all these equivalent theorems...

Answer 6

I can see that the transpose of the matrix does have a pivot in every column , i just dont understand what the bookmeans by linear independence

Answer 7

linear independent means what you said; that no vector in the set can be written as the sum of a scalar multiples of other vectors in the set.
However there are a number of other theorem that are implied by that. Here's a decent list of many.
http://tutorial.math.lamar.edu/Classes/LinAlg/FundamentalSubspaces.aspx

Answer 8

Theorem 8

Answer 9

okay - as dependence would be the opposite .. hmm one more question: "if the matrix A is lin ind. if and only if the eq. Ax=0 has the only trivial sol'n"
What are they refering to by 'trivial sol'n' ??

Answer 10

x=0

Answer 11

to the homogeneous eq .. what does that imply what 'x' is ?
is the question

Answer 12

that the only solution for, say, a 3x3 matrix \(A\vec x=\vec 0\) is \[\vec x=\left[\begin{matrix}x\\y\\z\end{matrix}\right]\]is x=0, y=0, z=0, or\[\vec x=\vec 0\]

Answer 13

Much appreciated Turing

Answer 14

which is obviously the case for anything that is row-reducible to the identity matrix\[\left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right]\left[\begin{matrix}x\\y\\z\end{matrix}\right]=\left[\begin{matrix}0\\0\\0\end{matrix}\right]\]so you can see that these theorems are well-connected. In fact they are all if and only if statements; i.e. they all imply each other.

Answer 15

very welcome :)

Answer 16

okay , That seems to be an issue . I cant see the connections . but this will get me to re-evaulate my work . thank you

Answer 17

welcome again!