Identity matrix problem

ausmathgenius420

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Aug 5, 2021
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44
Hi,
My question is if you have 2 matrices A and B, such that:
AB=24IAB=24IHow can I prove that;
A1=124BA^{-1}=\frac{1}{24}B
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
AB=xIAB=xIThus
A1=1xBA^{-1}=\frac{1}{x}B
 
Last edited:
AB=xI\displaystyle AB = xI
A1AB=A1.xI\displaystyle A^{-1} A B = A^{-1}. xI
IB=xA1I\displaystyle IB = x A^{-1}I
B=xA1\displaystyle B = x A^{-1}
1xB=1x.xA1\displaystyle \frac{1}{x}B = \frac{1}{x}.x A^{-1}

A1=1xB\displaystyle A^{-1} = \frac{1}{x}B

(Can someone tell me how to line up the = signs using LaTex?)
 
Last edited:
AB=xI\displaystyle AB = xI
A1AB=A1.xI\displaystyle A^{-1} A B = A^{-1}. xI
IB=xA1I\displaystyle IB = x A^{-1}I
B=xA1\displaystyle B = x A^{-1}
1xB=1x.xA1\displaystyle \frac{1}{x}B = \frac{1}{x}.x A^{-1}

A1=1xB\displaystyle A^{-1} = \frac{1}{x}B

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