Can I use algebraic identities with matrices? Let $A$ and $B$ be square matrices, so that it is possible to add or multiply them with themselves and with each other.
Can algebraic identities such as $(A+B)^2=A^2+2AB+B^2$ apply to them?
 A: The algebra of matrices is not the same as the algebra of (real or complex) numbers. The two main differences are

*

*Any two numbers can be multiplied, but only matrices of "compatible dimension" can be multiplied


*The product of any two numbers is independent of the order, but only some pair of matrices have this property
If you limit yourself to square matrices, you can eliminate the first issue, but they still may not commute in general.
This is a great motivation for understanding proofs. If you can carefully prove that $(A+B)^2 = A^2 + 2AB + B^2$ for real numbers $A$ and $B$, you can repeat that proof for an algebra which is not commutative and see not only where it falls apart, but what the adjusted identity should be -- in this case, $(A+B)^2 = A^2 + AB + BA + B^2$.
As you advance in math, you will come across various different algebras that have different combinations of properties such as associativity, commutativity, invertibility, etc.; instead of memorizing a whole sheet of identities for each and every one, it would behoove you to be able to derive those properties from the ones given by definition. An introductory text on abstract algebra would be a great place to start if you're interested in learning more.
A: Matrix multiplication is not (in general) commutative.
So $(A+B)^2=(A+B)(A+B)=A^2+AB+BA+B^2$.
You can't (generally) combine the $AB$ and $BA$ terms, since this step would require $BA=AB$.
