Identity matrix problem

ausmathgenius420

New member
Joined
Aug 5, 2021
Messages
7
Hi,
My question is if you have 2 matrices A and B, such that:
[math]AB=24I[/math]How can I prove that;
[math]A^{-1}=\frac{1}{24}B[/math]
Intuitively it makes sense, however my textbook doesn't provide a proof for this which I'm curious about so hopefully someone can explain.

Edit:
24I was just an example I chose. These equations can be generalised to
[math]AB=xI[/math]Thus
[math]A^{-1}=\frac{1}{x}B[/math]
 
Last edited:

Harry_the_cat

Elite Member
Joined
Mar 16, 2016
Messages
2,977
\(\displaystyle AB = xI\)
\(\displaystyle A^{-1} A B = A^{-1}. xI\)
\(\displaystyle IB = x A^{-1}I\)
\(\displaystyle B = x A^{-1}\)
\(\displaystyle \frac{1}{x}B = \frac{1}{x}.x A^{-1}\)

\(\displaystyle A^{-1} = \frac{1}{x}B\)

(Can someone tell me how to line up the = signs using LaTex?)
 
Last edited:

ausmathgenius420

New member
Joined
Aug 5, 2021
Messages
7
\(\displaystyle AB = xI\)
\(\displaystyle A^{-1} A B = A^{-1}. xI\)
\(\displaystyle IB = x A^{-1}I\)
\(\displaystyle B = x A^{-1}\)
\(\displaystyle \frac{1}{x}B = \frac{1}{x}.x A^{-1}\)

\(\displaystyle A^{-1} = \frac{1}{x}B\)

(Can someone tell me how to line up the = signs using LaTex?)
Thank you!
 
Top