Question

State the conditions on the constants which A will be invertible? [k 0 0 0; 1 k 0 0 ; 0 1 k 0; 0 0 1 k]

Answer 1

from first glance, im gonna say that k isnt -1.

Answer 2

because if k was negative one, then the matrix equation Ax = 0 has a non trivial solution, the vector (1,1,1,1)

Answer 3

theres gotta be more to it though <.< ima think about it more lol

Answer 4

we're not necessarily looking for k? I know the answer but just don't know how they got it

Answer 5

may be you should use detA =/= 0

Answer 6

oops, i didnt look at the first row, disregard my answer

Answer 7

This section is before learning about determinants, it's in elementary matrices let me post the answer

Answer 8

isnt the determinant of that matrix k^4?

Answer 9

since it is triangular?

Answer 10

detA = k^4, so k=/= 0

Answer 11

right right :)

Answer 12

1/k^4 [k^2 0 0 0; -k^2 k^2 0 0; k -k^2 k^2 0; -1 k -k^2 k^2]

Answer 13

so they actually gave you the inverse of the matrix. thats what that guy looks like.

Answer 14

and you can see from the fraction on the outside that k^4 cant be 0 or we would be dividing by 0

Answer 15

so how do you go from the matrix to this answer?

Answer 16

you need to use row reduction

Answer 17

But neither of the matrices are row reduced?

Answer 18

The answer is the inverse correcT?

Answer 19

If that is the case you can find the inverse with row reduction.

Answer 20

The composition of the elementary operations required to take it to the identity is the inverse.

Answer 21

im trying to find that gross formula where you use the cofactors of the matrix, its something like:
\[A^{-1} = \frac{1}{detA}A_c\]
i dont like using it but it might be the best way to do this problem. do you know what im talking about Alchemista?

Answer 22

The simplest way to find the inverse is by setting up two matrices. One is the original matrix one is the identity.

Answer 23

As you perform row operations on the original matrix perform the same on the identity.

Answer 24

Gotcha yeah that's how I got it too now

Answer 25

When you reduce it to the identity the other matrix will be the inverse.

Answer 26

thanks for your help everyone

Answer 27

the adjugate matrix, using this:
http://en.wikipedia.org/wiki/Cofactor_(linear_algebra)

Answer 28

I guess that works too but I think the other way is simplest?

Answer 29

Yes I understand what you are saying but its not as efficient as row reduction. In fact calculating the det in some cases will be much worse than row reduction.

Answer 30

its a diagonal matrix >.< its gonna be really easy to get determinants i know this isnt efficient all the time, i hate this method, but the matrix calls for it