A property of matrix multiplication is that, if $A$ & $B$ & $C$ are matrices
- $A(BC) = (AB)C$ (associative)
- $A(B+C)$ = $AB + AC$ and $(A + B)C = AC + BC$ (distributive)
- $AB ≠ BA$ (not commutative)
And if $A$ is invertible and $AB=I$, then $BA=I$, hence B is the inverse of A, then:
$AB = I$
$A^{-1}AB = A^{-1}I$
$IB = A^{-1}I$
$B = A^{-1}$
$BA = A^{-1}A = I$
and I want to find X - an unknown matrix. This is true:
$AX = C $
$A^{-1} AX = A^{-1} C$
$IX = A^{-1}C$
$X = A^{-1} C$
But is the below variation also correct according to the matrix definitions and why? Does it matter where $A^{-1}$ is on the RHS, before or after $C$? Does it make a difference if $I$ is before or after $X$? The matrix definition says it is not commutative so the output is $IX ≠ XI$?
$XA = C$
$XAA^{-1} = CA^{-1}$
$XI = CA^{-1}$
$X = CA^{-1}$