Can you square both sides of an equation containing matrices?

Question

For example A=λI ⇒ A²=λ²I where A is square, and λ∈ℝ.
Or more generally AB=CD ⇒ ABAB=CDCD.
Assuming all the necessary matrix products are possible, what other conditions would need to be fulfilled for this to hold?

Answer 1

For my money, the answer you’re looking for has nothing whatever to do with matrices. For, in mathematics, when we write$$A=B$$ we’re saying that $A$ and $B$ are the same thing. In your case, you’re not talking about one matrix $AB$ and another matrix $CD$, rather you’re talking about a single matrix that has two factorizations.

Thus the principle that “you may always do something to one side of an equation as long as you do the same thing to the other” is not a mathematical principle, but rather a principle of logic, dependent on the mathematical meaning of equality.

Answer 2

$AB=CD$

$\Rightarrow ABAB=CDAB$

$\Rightarrow ABAB=CDCD$

In words: you are allowed to right multiply both sides by $AB$, which is the same as $CD$, so the answer is yes (given that you can actually square the matrix, i.e. that it is a square matrix).