[গণিত] নির্ণয় কর।
উত্তর: দেওয়া আছে, [গণিত] নির্ণয়যোগ্য। [গণিত] B এর সহগুণকসমূহ : [গণিত] নির্ণেয় [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
[গণিত] নির্ণয় কর।
উদ্দীপক
(i) \left|\[\begin{array}{ccc}a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\]\right|-A=0.
(ii) \mathbf{B}=\left[\[\begin{array}{rrr}3 & -4 & 2 \\ -2 & 1 & 0 \\ -1 & -4 & 3\end{array}\]\right].
উত্তর
বামপক্ষ =\left|\[\begin{array}{ccc} 3 & p & 6-p \\ 5 & q & 10-q \\ 7 & r & 14-r \end{array}\]\right|
=\left|\[\begin{array}{ccc} 3 & p & 6-6+p \\ 5 & q & 10-10+q \\ 7 & r & 14-14+r \end{array}\]\right| \quad\left[c_3{ }^{\prime}=2 c_1-c_3\right]
=\left|\[\begin{array}{lll} 3 & p & p \\ 5 & q & q \\ 7 & r & r \end{array}\]\right|
শেষ দুইটি কলাম একই]
= ডানপক্ষ
\therefore\left|\[\begin{array}{ccc} 3 & p & 6-p \\ 5 & q & 10-q \\ 7 & r & 14-r \end{array}\]\right|=0 (দেখানো হলো)
উত্তর
দেওয়া আছে, \left|\[\begin{array}{ccc}a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\]\right|-A=0
বা, A=\left|\[\begin{array}{ccc}a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\]\right|
=\left|\[\begin{array}{ccc}a+b+c & a+b+c & a+b+c \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\]\right| \quad\left[r_1^{\prime}=r_1+r_2+r_3\right]
=(a+b+c)\left|\[\begin{array}{ccc}1 & 1 & 1 \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\]\right|
=(a+b+c)\left|\[\begin{array}{ccc}0 & 0 & 1 \a+b+c & -(a+b+c) & 2 b \\ 0 & a+b+c & c-a-b\end{array}\]\right|\left[\[\begin{array}{l}c_1^{\prime}=c_1-c_2, \c_2^{\prime}=c_2-c_3\end{array}\]\right]
=(a+b+c)\left|\[\begin{array}{cc}a+b+c & -(a+b+c) \\ 0 & a+b+c\end{array}\]\right|
(দেখানো হলো)
উত্তর
দেওয়া আছে, \mathrm{B}=\left[\[\begin{array}{rrr}3 & -4 & 2 \\ -2 & 1 & 0 \\ -1 & -4 & 3\end{array}\]\right]
\therefore|B| =\left|\[\begin{array}{rrr} 3 & -4 & 2 \\ -2 & 1 & 0 \\ -1 & -4 & 3 \end{array}\]\right|
নির্ণয়যোগ্য।
Adj B=\left[\[\begin{array}{lll}\mathbf{B}_{11} & \mathbf{B}_{12} & \mathbf{B}_{13} \\ \mathbf{B}_{21} & \mathbf{B}_{22} & \mathbf{B}_{23} \\ \mathbf{B}_{31} & \mathbf{B}_{32} & \mathbf{B}_{33}\end{array}\]\right]^{\mathbf{T}}
B এর সহগুণকসমূহ :
\mathrm{B}_{11}=\left|\[\begin{array}{rr} 1 & 0 \\ -4 & 3 \end{array}\]\right|=3+0=3
\mathrm{~B}_{12}=-\left|\[\begin{array}{ll} -2 & 0 \\ -1 & 3 \end{array}\]\right|=-(-6+0)=6
\mathrm{~B}_{13}=\left|\[\begin{array}{rr} -2 & 1 \\ -1 & -4 \end{array}\]\right|=8+1=9
\mathrm{~B}_{21}=-\left|\[\begin{array}{rr} -4 & 2 \\ -4 & 3 \end{array}\]\right|=-(-12+8)=4
\mathrm{~B}_{22}=\left|\[\begin{array}{rr} 3 & 2 \\ -1 & 3 \end{array}\]\right|=9+2=11
\mathrm{~B}_{23}=-\left|\[\begin{array}{rr} 3 & -4 \\ -1 & -4 \end{array}\]\right|=-(-12-4)=16
\mathrm{~B}_{31}=\left|\[\begin{array}{rr} -4 & 2 \\ 1 & 0 \end{array}\]\right|=0-2=-2 ; \mathrm{B}_{32}
=-\left|\[\begin{array}{rr} 3 & 2 \\ -2 & 0 \end{array}\]\right|=-(0+4)=-4
\mathrm{B}_{33}=\left|\[\begin{array}{rr}3 & -4 \\ -2 & 1\end{array}\]\right|=3-8=-5
\therefore \quad \mathrm{AdjB}=\left[\[\begin{array}{rrr}3 & 6 & 9 \\ 4 & 11 & 16 \\ -2 & -4 & -5\end{array}\]\right]^{\mathrm{T}}
=\left[\[\begin{array}{lll}3 & 4 & -2 \\ 6 & 11 & -4 \\ 9 & 16 & -5\end{array}\]\right]
\therefore \quad \mathrm{B}^{-1}=\frac{1}{|\mathrm{~B}|} \mathrm{AdjB}=\frac{1}{3}\left[\[\begin{array}{rrr}3 & 4 & -2 \\ 6 & 11 & -4 \\ 9 & 16 & -5\end{array}\]\right]
নির্ণেয় \mathrm{B}^{-1}=\frac{1}{3}\left[\[\begin{array}{rrr}3 & 4 & -2 \\ 6 & 11 & -4 \\ 9 & 16 & -5\end{array}\]\right] .