Chapter 3.6, Problem 3P

In Problems 1 to 3, find $AB,BA,A+B,A-B,A^2,B^2,5A,3B$.

Observe that $AB\ne BA$.

Show that $(A-B)(A+B)\ne (A+B)(A-B)=A^2-B^2$.

Show that det AB det BA = (det A)(det B), but that $\det (A+B)\ne \det A+\det B$.

Show that $\det (5A)\ne 5\text{}\det A$, and find n so that $\det (5A)=5^n\det A$.

Find similar results for det(3B). Remember that the point of doing these simple problems by hand is to learn how to manipulate determinants and matrices correctly. Check your answers by computer.

$A=\left(\begin{array}{l}102\\ 3-10\\ 051\end{array}\right)$, $B=\left(\begin{array}{l}110\\ 021\\ 3-10\end{array}\right)$