Say I have the following relationship between matrices:
$AB = A^{2} + 2A$
If I multiply both sides of the equation by $A^{-1}$ is the resulting equation equivalent, meaning it doesn't change the values of A or B? So for example:
$(A^{-1})AB = (A^{-1})A^{2} + (A^{-1})(2A)$
And thus get:
$B = A + 2I$
Is this correct?
EDIT: Assume A is a square invertible matrix.
Best Answer
Provided an inverse exists multiplying by an inverse matrix preserves the relation you started with. This is analogous to dividing two sides of a relation. For example if $x,y\in\mathbb{R}$:
$xy=x^{2}+2x$
Provided that $x\neq0$ then:
$\frac{xy}{x}=\frac{x^{2}}{x}+\frac{2x}{x}$
$y=x+2$.