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Matrices Q FP1

A-1C-1 = (CA)-1

Please can someone explain why?

Thanks in advance
Original post by tsr13579
A-1C-1 = (CA)-1

Please can someone explain why?

Thanks in advance


Let X=A−1C−1X = A^{-1}C^{-1}

Front multiply by A.

Front multiply by C.
Reply 2
Avatar for tsr13579
tsr13579
OP
let X = A-1C-1

Front multiply by A: AX = A-1AC-1 = IC = C
Front multiply by C: CAX = CC-1 = I

I've ended up with CAX=I, where do I go from here?
Reply 3
Avatar for Phichi
Phichi
11
Original post by tsr13579
let X = A-1C-1

Front multiply by A: AX = A-1AC-1 = IC = C
Front multiply by C: CAX = CC-1 = I

I've ended up with CAX=I, where do I go from here?


If you let Y=CAY = CA

Then Y−1CAX=Y−1Y^{-1}CAX = Y^{-1}

As Y−1=(CA)−1Y^{-1} = (CA)^{-1}

X=(CA)−1 X = (CA)^{-1}
Original post by tsr13579
let X = A-1C-1

Front multiply by A: AX = A-1AC-1 = IC = C
Front multiply by C: CAX = CC-1 = I

I've ended up with CAX=I, where do I go from here?


Slightly worried by this!

You should know that PP−1=IP P^{-1} = I so X=(CA)−1X=(CA)^{-1}

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