First Principles AI

Solution

In ordinary algebra, we often expand using commutativity (i.e. $ab=ba$ ). For matrices, in general:

AB\ne BA

So expansions that rely on combining $AB$ and $BA$ are not valid.

Expand:

(A+B)^2=(A+B)(A+B)=A^2+AB+BA+B^2

For it to equal $A^2+2AB+B^2$ , we would need $AB+BA=2AB$ , i.e. $BA=AB$ , which is not true in general.

Hence:

\boxed{(A+B)^2\ne A^2+2AB+B^2\ \text{(in general)}}

Expand:

(A-B)^2=(A-B)(A-B)=A^2-AB-BA+B^2

For it to equal $A^2-2AB+B^2$ , we would need $-AB-BA=-2AB$ , i.e. $BA=AB$ , which is not true in general.

So:

\boxed{(A-B)^2\ne A^2-2AB+B^2\ \text{(in general)}}

Expand:

(A+B)(A-B)=A^2-AB+BA-B^2

For it to equal $A^2-B^2$ , we would need $-AB+BA=0$ , i.e. $BA=AB$ , which is not true in general.

Thus:

\boxed{(A+B)(A-B)\ne A^2-B^2\ \text{(in general)}}