দেখাও যে, [গণিত] .
উত্তর: দেওয়া আছে, [গণিত] [ [গণিত] এবং [গণিত] প্রয়োগ করে] [গণিত]
HSC Higher Math 1st Paper Chapter 1: ম্যাট্রিকস ও নির্ণায়ক CQ — Prosthuti প্রশ্নব্যাংক
HSC · Higher Math 1st Paper · Chapter 1: ম্যাট্রিকস ও নির্ণায়ক · সৃজনশীল
দেখাও যে, .
উদ্দীপক
A=\left[\[\begin{array}{cc}4 & 1 \\ 3 & M\end{array}\]\right], B=\left[\[\begin{array}{ccc}p & q & r \p^{2} & q^{2} & r^{2} \p^{3}-1 & q^{3}-1 & r^{3}-1\end{array}\]\right] এবং C=\left[\[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\]\right]
Chittagong · 2021
উত্তর
আমরা জানি, ম্যাট্রিক্স এর মুখ্যকর্ণের যোগফলকে ট্রেস বলে।
\therefore\left[\[\begin{array}{ll}4 & 1 \\ 3 & M\end{array}\]\right] এর ট্রেস
প্রশ্নমতে,
বা,
উত্তর
দেওয়া আছে, B=\left[\[\begin{array}{ccc}p & q & r \p^{2} & q^{2} & r^{2} \p^{3}-1 & q^{3}-1 & r^{3}-1\end{array}\]\right]
\therefore|B|=\left|\[\begin{array}{ccc}p & q & r \p^{2} & q^{2} & r^{2} \p^{3}-1 & q^{3}-1 & r^{3}-1\end{array}\]\right|
=\left|\[\begin{array}{lll}p & q & r \p^{2} & q^{2} & r^{2} \p^{3} & q^{3} & r^{3}\end{array}\]\right|-\left|\[\begin{array}{ccc}p & q & r \p^{2} & q^{2} & r^{2} \\ 1 & 1 & 1\end{array}\]\right|
=p q r\left|\[\begin{array}{ccc}1 & 1 & 1 \p & q & r \p^{2} & q^{2} & r^{2}\end{array}\]\right|-\left|\[\begin{array}{ccc}1 & 1 & 1 \p & q & r \p^{2} & q^{2} & r^{2}\end{array}\]\right|
=(p q r-1)\left|\[\begin{array}{lll}1 & 1 & 1 \p & q & r \p^{2} & q^{2} & r^{2}\end{array}\]\right|
=(p q r-1)\left|\[\begin{array}{ccc}0 & 0 & 1 \p-q & q-r & r \p^{2}-q^{2} & q^{2}-r^{2} & r^{2}\end{array}\]\right|
[ এবং প্রয়োগ করে]
=(p q r-1)\left|\[\begin{array}{ll}p-q & q-r \p^{2}-q^{2} & q^{2}-r^{2}\end{array}\]\right|
=(p q r-1)(p-q)(q-r)\left|\[\begin{array}{cc}1 & 1 \p+q & q+r\end{array}\]\right|
উত্তর
দেওয়া আছে, C=[ \left.\[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\]\right]
\therefore|C|=\left|\[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\]\right|
বিদ্যমান।
\operatorname{Adj} C=\left[\[\begin{array}{lll}C_{11} & C_{12} & C_{13} \C_{21} & C_{22} & C_{23} \C_{31} & C_{32} & C_{33}\end{array}\]\right]^{\top}
C এর সহগুণকসমূহ :
C_{11}=\left|\[\begin{array}{ll}2 & 3 \\ 1 & 1\end{array}\]\right|=2-3=-1: C_{12}=-\left|\[\begin{array}{ll}1 & 3 \\ 3 & 1\end{array}\]\right|=-(1-9)=8
C_{13}=\left|\[\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\]\right|=1-6=-5 ; C_{21}=-\left|\[\begin{array}{ll}1 & 2 \\ 1 & 1\end{array}\]\right|=-(1-2)=1
C_{22}=\left|\[\begin{array}{ll}0 & 2 \\ 3 & 1\end{array}\]\right|=0-6=-6 ; C_{23}=-\left|\[\begin{array}{ll}0 & 1 \\ 3 & 1\end{array}\]\right|=-(0-3)=3
C_{31}=\left|\[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\]\right|=3-4=-1 ; C_{32}=-\left|\[\begin{array}{ll}0 & 2 \\ 1 & 3\end{array}\]\right|=-(0-2)=2
\mathrm{C}_{33}=\left|\[\begin{array}{ll}0 & 1 \\ 1 & 2\end{array}\]\right|=0-1=-1
Adj C=\left[\[\begin{array}{rrr}-1 & 8 & -5 \\ 1 & -6 & 3 \\ -1 & 2 & -1\end{array}\]\right]^{T}=\left[\[\begin{array}{rrr}-1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1\end{array}\]\right]
. Adj C=\frac{1}{-2}\left[\[\begin{array}{rrr}-1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1\end{array}\]\right]
=\frac{1}{2}\left[\[\begin{array}{rrr}1 & -1 & 1 \\ -8 & 6 & -2 \\ 5 & -3 & 1\end{array}\]\right]
নির্ণেয় C^{-1}=\frac{1}{2}\left[\[\begin{array}{rrr}1 & -1 & 1 \\ -8 & 6 & -2 \\ 5 & -3 & 1\end{array}\]\right]