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Inverse Matrices

Hi I was wondering if anyone could help me with this:

Using the definition of an inverse matrix and the principles of matrix algebra, prove that if A and B are invertible matrices of the same order, then AB is invertible and (AB)^-1=B^-1A^-1.
I'll start you off:

X=(AB)−1X = (AB)^{-1}

  ⟺  (AB)X=I\iff (AB) X = I

  ⟺  A(BX)=I\iff A (BX) = I

  ⟺  A−1A(BX)=A−1I\iff A^{-1} A (BX) = A^{-1} I

  ⟺  BX=A−1\iff BX = A^{-1}

Now keep going in that vein.
Mr M
I'll start you off:

X=(AB)−1X = (AB)^{-1}

  ⟺  (AB)X=I\iff (AB) X = I

  ⟺  A(BX)=I\iff A (BX) = I

  ⟺  A−1A(BX)=A−1I\iff A^{-1} A (BX) = A^{-1} I

  ⟺  BX=A−1\iff BX = A^{-1}

Now keep going in that vein.



X=A^-1 X 1/B

X=A^-1.B^-1

X=(AB)^-1

so this shows that AB is invertible and I've interchanged A and B in the second line and managed to get (AB)^-1=B^-1.A^-1 as well.

Cheers. :smile:

Btw any latex software that you recommend?
The Dark Side
X=A^-1 X 1/B

X=A^-1.B^-1

X=(AB)^-1

so this shows that AB is invertible and I've interchanged A and B in the second line and managed to get (AB)^-1=B^-1.A^-1 as well.

Cheers. :smile:

Btw any latex software that you recommend?

excuse me?
Nope I only started using it about a month ago. If you want to have a go at LaTeX, edit your previous post and you will be able to see exactly what I typed to produce mine and you can amend yours accordingly.

And you need to work on that again. You can't "divide" matrices like that. Multiply by B−1B^{-1}. Remember the order of multiplication is important.
Reply 5
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wellison999
The Dark Side
X=A^-1 X 1/B


Careful there - 1/B doesn't really have meaning for matrices - this is why we have to 'multiply' by B−1 B^{-1} to cancel a B. Also note that it makes a difference if you multiply from the right or left.

What it should be is:

B−1BX=B−1A−1 B^{-1} B X = B^{-1} A^{-1}

X=B−1A−1 X = B^{-1} A^{-1}


Btw any latex software that you recommend?


I use BaKoMa TeX (see http://bakoma-tex.com/) - as does the rest of the Imperial maths department. Quite simple to use really.
Sorry I am new to matrices. I wrote 1/B but I meant B^-1.
Dark Side,

I was more concerned about where you put it. It needs to be at the front.
BXB−1=A−1B−1BXB^{-1}=A^{-1}B^{-1}

but how does this get rid of the first B?...
Totally Tom
BXB−1=A−1B−1BXB^{-1}=A^{-1}B^{-1}

but how does this get rid of the first B?...



????
The BB and the B−1B^{-1} have to be side by side before they "cancel out".

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