Question $5$

Suppose the function $($:$A,B$:$)$ is defined over matrices $A,Bin{R}_{mn}$

as follows,

$($:$A,B$:$)={\displaystyle \underset{i=1}{\overset{m}{}}}{\displaystyle \underset{j=1}{\overset{n}{}}}{a}_{ij}{b}_{ij}$

$=a_{11}{b}_{11}+a_{12}{b}_{12}+$cdots$+a_{1n}{b}_{1n}$

$+a_{21}{b}_{21}+a_{22}{b}_{22}+$cdots$+a_{2n}{b}_{2n}$

$+$cdots

$+a_{m1}{b}_{m1}+a_{m2}{b}_{m2}+$cdots$+a_{mn}{b}_{mn}$

For example, if $A=\left[\begin{array}{ccc}2& -1& 3\\ 5& 0& 4\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 3& 8\\ 0& 1& -2\end{array}\right],$ then

$($:$A,B$:$)=(2)(1)+(-1)(3)+(3)(8)+(5)(0)+(0)(1)+(4)(-2)=$

Select all the properties that $($:$A,B$:$)$ satisfies. Assume all given

matrices are in $R_{mn}.$

$($:$A,Z$:$)=0$ where $Zin{R}_{mn}$ and $z_{ij}=0,$AAi,$j$

$($:$A,B$:$)=($:$B,A$:$)$

$($:$A,B$:$)=($:$A,B$:$)$ for real number $$

$($:$A,A$:$)>0$ if $A$ is not a matrix of just $0$ entries.

$($:$A+B,C$:$)=($:$A,C$:$)+($:$B,C$:$)$