If we have $2$ different Matrices $A$ and $B$, is $(AB)^2 $ equals to $A^2 B^2$??

Question

Does $(A B)^2$ equals to $A^2 B^2$ in Matrices???? Please explain your answer. Thank you!! Please help me

Answer 1

In general, they are not the same \begin{align} (AB)^{2} \;=\; (AB) (AB) \; = \; ABAB \; \neq \; A^{2} B^{2} \end{align} But if we assume $AB = BA$ (i.e. $A$ and $B$ are commute), then we have \begin{align} (AB)^{2} \; = \; A(BA)B \; = \; A(AB)B \; = \; A^{2}B^{2} \end{align}

Answer 2

for a given matrix $A$, $A^2$ means you take the matrix $A$ and you multiply it by itself. So $(AB)^2 = (AB \cdot AB) = (ABAB)$, but as I'm sure you've experienced, matrix multiplication is not commutative (i.e. $AB \not = BA$ all the time) So $(AB)^2 \not = A^2\cdot B^2$.

As another note, in order to square a matrix, you need it to have the same number of columns as it has rows. If $A$ is a 2 x 3 matrix, and $B$ is a 3 x 2 matrix, then $(AB)$ will be a 2 x 2 matrix so I can square it, (I have the same number of rows and columns) but neither $A^2$ nor $B^2$ exist! (Neither of them have the same number of rows as they have columns) So in general, $(AB)^2 \not = A^2 \cdot B^2$.