সৃজনশীলঃ ‘দৃশ্যকল্প -১: A=[1364],B=[2143]দৃশ্যকল্প…’ দিনাজপুর বোর্ড ২০১৯ – ম্যাট্রিকস ও নির্ণায়ক, উচ্চতর গণিত, উচ্চতর গণিত ১ম পত্র (এইচএসসি)

দেওয়া আছে ,

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

$C=A-B$

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

$=\left[\begin{array}{cccc}1& -2& 3& -1\\ 6& -4& 4& -3\end{array}\right]$

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

এখন, $C^2=\left[\begin{array}{cc}-1& 2\\ 2& 1\end{array}\right]\left[\begin{array}{cc}-1& 2\\ 2& 1\end{array}\right]$

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

$=\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$

$\therefore C^2+5B+3I=\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]+5\left[\begin{array}{cc}2& 1\\ 4& 3\end{array}\right]+3\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$=\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]+\left[\begin{array}{cc}10& 5\\ 20& 15\end{array}\right]+\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$

$=\left[\begin{array}{cc}5+10+3& 0+5+0\\ 0+20+0& 5+15+3\end{array}\right]$

$=\left[\begin{array}{cc}18& 5\\ 20& 23\end{array}\right]$

দৃশ্যকল্প -২ এ প্রদত্ত সমীকরণ জোট,

$2x+3y-5z=7$

$x-4y+z=4$

$\frac{3}{5}x-\frac{1}{5}y-\frac{2}{5}z=1$

চলকসমূহের সহগগুলি নিয়ে,

$D=\left|\begin{array}{ccc}2& 3& -5\\ 1& -4& 1\\ \frac{3}{5}& -\frac{1}{5}& -\frac{2}{5}\end{array}\right|$

$=2\times (\frac{8}{5}+\frac{1}{5})-3(-\frac{2}{5}-\frac{3}{5})-5(-\frac{1}{5}+\frac{12}{5})$

$=\frac{18}{5}+3-11$

$=-\frac{22}{5}$

$D_x=\left[\begin{array}{ccc}7& 3& -5\\ 4& -4& 1\\ 1& -\frac{1}{5}& -\frac{2}{5}\end{array}\right]$

$=7(\frac{8}{5}+\frac{1}{5})-3(-\frac{8}{5}-1)-5(-\frac{4}{5}+4)$

$=\frac{63}{5}+\frac{39}{5}-16$

$=\frac{22}{5}$

$D_y=\left|\begin{array}{ccc}2& 7& -5\\ 1& 4& 1\\ \frac{3}{5}& 1& -\frac{2}{5}\end{array}\right|$

$=2(-\frac{8}{5}-1)-7(-\frac{2}{3}-\frac{3}{5})-5(1-\frac{12}{5})$

$=-\frac{26}{5}+7+7$

$=\frac{44}{5}$

$D_z=\left|\begin{array}{ccc}2& 3& 7\\ \frac{1}{3}& -4& 4\\ \frac{3}{5}& -\frac{1}{5}& 1\end{array}\right|$

$=2(-4+\frac{4}{5})-3(1-\frac{12}{5})+7(-\frac{1}{5}+\frac{12}{5})$

$=-\frac{32}{5}+\frac{21}{5}+\frac{77}{5}$

$=\frac{66}{5}$

$\therefore x=\frac{D_x}{D}$

$=\frac{\frac{22}{5}}{\frac{-22}{5}}$

$=-1$

$y=\frac{D_y}{D}$

$=\frac{\frac{44}{5}}{\frac{-22}{5}}$

$=-2$

এবং $z=\frac{D_x}{D}$

$=\frac{D_z}{D}$

$=\frac{\frac{66}{5}}{\frac{-22}{5}}$

$=-3$

নির্ণেয সমাধান , $(x,y,z)=(-1,-2,-3)$