Question

I'm working on finding a formula for (A+B)^2 (square of sum of 2 matrices). I compute A^2, B^2, and AxB, and I also square the sum A+B, but I can't find a relationship between them. I'm working on problem set 2.4, #6. Any hints for finding the correct rule?

Answer 1

Well, here are a couple of hints (you are SO close :). You are missing one piece of the puzzle. \[A^2+B^2+AB+\underline{\;\;\;}=\left(A+B\right)^2 \]

Think about squaring the sum of two general matrices\[A_{2x2}+B_{2x2}\]
\[A+B = \left[\begin{matrix} a_{11}+b_{11}&a_{12}+b_{12}\\ a_{21}+b_{21}&a_{22}+b_{22} \end{matrix}\right] \]
So what would be the spot in row 1, column 2 of \[(A+B)^2\] look like? It would be
\[a_{11}*a_{12}+....\]Once you work that out, you'll probably recognize what the overall formula is.

Answer 2

Also, kudos for asking for hints and not the solution outright.

Answer 3

I think I've found a solution using your method but will want to check it when I am more awake. Thanks.

Answer 4

Two bacis rules are A*B≠B*A,and (A+B)*C=A*C+B*C.
So,we get (A+B)*(A+B)=(A+B)*A+(A+B)*B=A^2+B*A+A*B+B^2.

In the real number system,a*b=b*a,so,(a+b)^=a^2+2ab+b^2,but,A,B are matrices.