Matrix Multiplication $(A-B)(A+B)$ For the matrix multiplication of $(A-B)(A+B)$, we get = $A^2 + AB - BA\ldots$?
I'm confused with how to deal with the last part of the equation. What is the correct way, for matrix multiplication, to deal with $-BB$?
Thanks.
 A: Dealing with the term $-BB$ is really the same thing as computing $(-1)BB = (-1)B^2$. If you are comfortable multiplying matrices, then this is nothing you haven't seen before.
With respect to Robert Israel's comment, for a scalar $s$, and a matrix $A$, the matrix formed by multiplying each entry of $A$ by $s$ is denoted $sA$. Thus, when we perform matrix multiplication $sAB$, this is the same as computing $AB$ and then multiplying the resulting matrix by $s$, or first computing $sA$ and then $(sA)B$, or first computing $sB$ and then computing $A(sB)$. Since all are equal we write
$$
sAB = (sA)B = A(sB).
$$
Let's think of matrix multiplication as a function that takes as input two matrices $A$ and $B$ and returns their product $AB$. We'll write this function with infix notation like this: $(\cdot,\cdot).$ Thus,
$$
(A,B) = AB.
$$
As Robert Israel pointed out, this is a bilinear function. That means that the function is linear in each of its two slots. So,
$$
(sA,B) = s(A,B) = (A,sB) = sAB
$$
and
$$
(A + B,C) = (A,C) + (B,C) = AC + BC \\
(A, B + C) = (A,B) + (A,C) = AB + AC.
$$
In general, a function $f$ of one input is linear if
$$
f(x + y) = f(x) + f(y) \\
f(\alpha x) = \alpha f(x).
$$
So when a function takes two inputs and each of its input slots behaves like a single-input linear function, that function is said to be bilinear.
Thus, we say that matrix multiplication is a bilinear operation.
A: $$(A-B)(A+B)=A^2+AB-BA-B^2$$
It's important to note that with matrices, it may not be the case that $AB = BA$, so we cannot simplify this expression any further.
