Question

If A and B are two matrices of same order $3\times 3$ where A= $\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 5& 6& 7\end{array}\right]$ and B= $\left[\begin{array}{ccc}3& 2& 5\\ 2& 3& 8\\ 7& 2& 9\end{array}\right]$ AB=?

Answer 1

$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 5& 6& 7\end{array}\right],B=\left[\begin{array}{ccc}3& 2& 5\\ 2& 3& 8\\ 7& 2& 9\end{array}\right]$
$AB=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 5& 6& 7\end{array}\right]\left[\begin{array}{ccc}3& 2& 5\\ 2& 3& 8\\ 7& 2& 9\end{array}\right]$
$=(a-b)(b-c)(c-a)[+\{ab-c^2-c(-a-b-c)\}]$
$=+(a-b)(b-c)(c-a)(ab+bc+ca)$
R.H.S=
$\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|$
$R_1\leftrightarrow R_1-R_2,R_2\leftrightarrow R_2-R_3$
$=\left|\begin{array}{ccc}0& a^2-b^2& a^3-b^3\\ 0& b^2-c^2& b^3-c^3\\ 1& c^2& c^3\end{array}\right|$
$=\left|\begin{array}{ccc}0& (a-b)(a+b)& (a-b)(a^2+ab+b^2)\\ 0& (b-c)(b+c)& (b-c)(b^2+bc+c^2)\\ 1& c^2& c^3\end{array}\right|$
$=(a-b)(b-c)\left|\begin{array}{ccc}0& a+b& a^2+ab+b^2\\ 0& b+c& b^2+bc+c^2\\ 1& c^2& c^3\end{array}\right|$
$R_1\leftrightarrow R_1-R_3$
$=(a-b)(b-c)\left|\begin{array}{ccc}0& (a+b)-(b+c)& (a^2+ab+b^2)-(b^2+bc+c^2)\\ 0& b+c& b^2+bc+c^2\\ 1& c^2& c^3\end{array}\right|$
$=(a-b)(b-c)\left|\begin{array}{ccc}0& a-c& (a-c)(a+c)+b(a-c)\\ 0& b+c& b^2+bc+c^2\\ 1& c^2& c^3\end{array}\right|$
$=(a-b)(b-c)(a-c)\left|\begin{array}{ccc}0& 1& a+b+c\\ 0& b+c& b^2+c^2+bc\\ 1& c^2& c^3\end{array}\right|$
$=\left[\begin{array}{ccc}1\times 3+2\times 2+3\times 7& 1\times 2+2\times 3+3\times 2& 1\times 5+2\times 8+3\times 9\\ 2\times 3+3\times 2+4\times 7& 2\times 2+3\times 3+4\times 2& 2\times 5+3\times 8+4\times 9\\ 5\times 3+6\times 2+7\times 7& 5\times 2+6\times 3+7\times 2& 5\times 5+6\times 8+7\times 9\end{array}\right]$
$={\left[\begin{array}{ccc}28& 14& 48\\ 40& 21& 70\\ 76& 42& 136\end{array}\right]}_{3\times 3}$