সৃজনশীলঃ ‘A=[4301],B=[14n0p] এবং C=[012120204]…’ কুমিল্লা বোর্ড ২০২১ – ম্যাট্রিকস ও নির্ণায়ক, উচ্চতর গণিত, উচ্চতর গণিত ১ম পত্র (এইচএসসি)

দেওয়া আছে,

$A=\left[\begin{array}{cc}4& 3\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}\frac{1}{4}& n\\ 0& p\end{array}\right]$

$\mathrm{AB}=\left[\begin{array}{cc}4& 3\\ 0& 1\end{array}\right]\left[\begin{array}{cc}\frac{1}{4}& n\\ 0& p\end{array}\right]$

$=\left[\begin{array}{cc}1& 4n+3p\\ 0& p\end{array}\right]$

প্রশ্নমতে,

$\mathrm{AB}=I$

$\left[\begin{array}{cc}1& 4n+3p\\ 0& p\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

উভয়পক্ষকে তুলনা করে পাই,

$4n+3p=0$

$\Rightarrow 4n=-3P$

$\Rightarrow n=\frac{-3}{4}p-\left(i\right)$

এবং $p=1$

P এর মান (i) নং এ বসিয়ে,

$n=\frac{-3}{4}$

$\therefore n=\frac{-3}{4},P=1$

দেওয়া আছে, $C=[\begin{array}{ccc}0& 1& 2\\ 1& 2& 0\\ 2& 0& 4\end{array}]$

$\therefore |C|=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 0\\ 2& 0& 4\end{array}\right]$

$=-1(4-0)+2(0-4)$

$=-4-8$

$=-12$

$\therefore C^{-1}$বিদ্যমান

$C_{11}=\left[\begin{array}{cc}2& 0\\ 0& 4\end{array}\right]C_{12}=\left[\begin{array}{cc}1& 0\\ 2& 4\end{array}\right]C_{13}=\left[\begin{array}{cc}1& 2\\ 2& 0\end{array}\right]$

$=(\begin{array}{cc}8-0)& =-(4-0)\end{array}=(0-4)$

$\begin{array}{ccc}=8& =-4& =-4\end{array}$

$C_{21}=\left[\begin{array}{cc}1& 2\\ 0& 4\end{array}\right]C_{22}=\left[\begin{array}{cc}0& 2\\ 2& 4\end{array}\right]C_{23}=\left[\begin{array}{cc}0& 1\\ 2& 0\end{array}\right]$

$=-(4-0)=(0-4)=-(0-2)$

$=-4=-4=2$

$C_{31}=\left[\begin{array}{cc}1& 2\\ 2& 0\end{array}\right]C_{12}=\left[\begin{array}{cc}1& 0\\ 2& 4\end{array}\right]C_{13}=\left[\begin{array}{cc}1& 2\\ 2& 0\end{array}\right]$

$=(0-4)=-(0-2)=(0-1)$

$\begin{array}{ccc}=-4& =2& =-1\end{array}$

$=\mathrm{Adj}C={\left[\begin{array}{ccc}8& -4& -4\\ -4& -4& 2\\ -4& 2& -1\end{array}\right]}^t$

$=\left[\begin{array}{ccc}8& -4& -4\\ -4& -4& 2\\ -4& 2& -1\end{array}\right]$

$\therefore C^{-1}=\frac{1}{\left|C\right|}\mathrm{adj}C=\frac{1}{-12}\left[\begin{array}{ccc}8& -4& -4\\ -4& -4& 2\\ -4& 2& -1\end{array}\right]$