Question

Another Linear algebra question.... see attachments please

Answer 1

First one is the question, the second is what I did, but I don't feel like I have accomplished anything, or if I did I am not sure if I showed it properly.

Answer 2

it is always true (matrices or whatever) that \[(AB)^{-1}=B^{-1}A^{-1}\]

Answer 3

ok

Answer 4

so \[B(AB)^{-1}=BB^{-1}A^{-1}=A^{-1}\]

Answer 5

I have that on my work

Answer 6

Did I do something wrong?

Answer 7

ok what are you trying to check?

Answer 8

I went to check to see if the A^-1 I got was correct and the work I did doesn't seem to match. Could you look at the document I uploaded?

Answer 9

you do not know what A is right? so there is no way to find \(A^{-1}\) other than to compute
\[B(AB)^{-1}\]

Answer 10

That's what I did

Answer 11

i guess i am confused
how are you trying to check your answer?

Answer 12

Are you able to see the picture I uploaded?

Answer 13

yeah
i checked your first multiplication, it is correct

Answer 14

OOOH i see the mistake

Answer 15

I checked by multiplying B^(-1)A^(-1) to see if I got back (AB)^-1

Answer 16

your determinant is wrong

Answer 17

Which one?

Answer 18

should be \(16-28\) not \(16-14\)

Answer 19

wow

Answer 20

Your right....

Answer 21

actually no, i am wrong

Answer 22

it is the determinant of the original matrix hold on

Answer 23

lol no i am right

Answer 24

that is the determinant of the original matrix sorry, still should be \(16-28\)

Answer 25

Is what I did to check correct? (Not the numbers, I know they are wrong, but is the method correct?)

Answer 26

yeah that should work
you are checking that
\[(AB)^{-1}=A^{-1}B^{-1}\] it should work out

Answer 27

And The question kind of confused me.... they ask me to find the inverse of A "If it is Possible". My Professor is big on Proofs. How do I show this "IS" possible?

Answer 28

All I know is that if (AB)^-1 exists than B^-1 and A^-1 should exist right?

Answer 29

i don't think you need to bother to check
the "if possible" business is because it is always possible that the inverse does not exist (if the determinant is zero) but in this case it does exist

Answer 30

yes you are right

Answer 31

Let me ask you this.... the only example I have been given to check if an inverse exists is in 2D..... would I check to see if the inverse of a 3x3 exists in the same way? Take the determinant of the 3x3, and be sure it doesn't equal 0? And if that is correct for a 3x3 is it correct for any mxn matrix as well?

Answer 32

you are told that \((AB)^{-1}\) exists, that means both \(A^{-1}\) and \(B^{-1}\) exist
if either one had determinant zero, then so would the product

Answer 33

to answer your question, yes, but if you are told the product has an inverse, then so must the factors

Answer 34

I get that, but I just wanted to know if the concept of a non-zero determinant implying the inverse exists is true in general for any size matrix/