Proof of R Transpose and Inverse Equals Identity Matrix

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The discussion centers on the proof that the transpose of a matrix R, denoted as RT, multiplied by R equals the identity matrix I, formally expressed as RTR = I. This implies that the inverse of R is equal to its transpose, R-1 = RT. A key point of confusion arises regarding the order of multiplication in matrix operations, specifically whether A A-1 = I can be rearranged to A-1A = I without altering the outcome.

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Looking over a proof of something, I was confused at this step:

We end up with: R[tex]^{T}[/tex] R = I (the identity matrix)
This must mean that R[tex]^{-1}[/tex] = R[tex]^{T}[/tex]

This is confusing me because I know that A A[tex]^{-1}[/tex] = I , but in the R case the transpose is on the left side instead of the right, and it seems to me that it matters what order the matrices are multiplied.
 
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Okay, start with [tex]AA^{-1}=I[/tex]. Can you turn this into [tex]A^{-1}A=I[/tex]?
 

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