Chapter 5.2, Problem 27E

Showing That a Function Is an Inner Product In Exercises 27 and 28, let

$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$ and $B=\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]$ be matrices in the vector space $M_{2,2}$. Show that the function defines an inner product on $M_{2,2}$.

$\langle A,B\rangle =a_{11}{b}_{11}+a_{12}{b}_{12}+a_{21}{b}_{21}+a_{22}{b}_{22}$