The monthly incomes of Aryan and Babban are in the ratio $3:4$ and their monthly expenditures are in the ratio $5:7$. If each saves $\mathrm{Rs} 15,000$ per month, find their monthly incomes using matrix method. This problem reflects which value?

Compute the indicated product:

 $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$

Compute the indicated product:

 $\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ -3& 2& -1\end{array}\right]$

Compute the indicated product:

$\left[\begin{array}{ccc}2& 3& 4\\ 3& 4& 5\\ 4& 5& 6\end{array}\right]\left[\begin{array}{ccc}1& -3& 5\\ 0& 2& 4\\ 3& 0& 5\end{array}\right]$

Show that $AB\ne BA$ in the following case:

 $A=\left[\begin{array}{cc}5& -1\\ 6& 7\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$

Show that $AB\ne BA$ in the following case:

 $A=\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]$

Show that $AB\ne BA$ in the following case:

 $A=\left[\begin{array}{ccc}1& 3& 0\\ 1& 1& 0\\ 4& 1& 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 5& 1\end{array}\right]$

Compute the products $AB$ and $BA$ if exists when $A=\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$

Compute the products $AB$ and $BA$ whichever exists when $A=\left[\begin{array}{cc}3& 2\\ -1& 0\\ -1& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}4& 5& 6\\ 0& 1& 2\end{array}\right]$