[গণিত] নির্ণেয় কর, যেখানে [গণিত] হচ্ছে A এর ট্রান্সপোজ ম্যাট্রিক্স।
উত্তর: [গণিত] ধরি, [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
নির্ণেয় কর, যেখানে হচ্ছে A এর ট্রান্সপোজ ম্যাট্রিক্স।
উদ্দীপক
A=\left[\[\begin{array}{ccc}4 & 1 & 5 \\ -2 & -1 & -3 \\ 3 & -4 & -9\end{array}\]\right], X=\left[\[\begin{array}{l}x \y \z\end{array}\]\right], B=\left[\[\begin{array}{l}2 \\ 5 \\ 4\end{array}\]\right]
এবং , যেখানে অভেদ ম্যাট্রিক্স
Jessore · 2021
উত্তর
মনে করি,
A=\left[\[\begin{array}{cc}2 & 3 \\ -1 & -2\end{array}\]\right]
A ম্যাট্রিক্সটি অভেদঘাতি হবে যদি হয়।
A^{2}=\left[\[\begin{array}{cr}2 & 3 \\ -1 & -2\end{array}\]\right]\left[\[\begin{array}{cr}2 & 3 \\ -1 & -2\end{array}\]\right]
=\left[\[\begin{array}{cc}4-3 & 6-6 \\ -2+2 & -3+4\end{array}\]\right]
=\left[\[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\]\right]=\mathrm{I}_{2}
A ম্যাট্রিক্সটি অভেদঘাতি।
উত্তর
A=\left[\[\begin{array}{ccc}4 & 1 & 5 \\ -2 & -1 & -3 \\ 3 & -4 & -9\end{array}\]\right] A^{\top}=\left[\[\begin{array}{ccc}4 & -2 & 3 \\ 1 & -1 & -4 \\ 5 & -3 & -9\end{array}\]\right]
ধরি,
P^{2}=P \times P=\left[\[\begin{array}{ccc}4 & -2 & 3 \\ 1 & -1 & -4 \\ 5 & -3 & -9\end{array}\]\right]\left[\[\begin{array}{ccc}4 & -2 & 3 \\ 1 & -1 & -4 \\ 5 & -3 & -9\end{array}\]\right]
=\left[\[\begin{array}{lll}16-2+15 & -8+2-9 & 12+8-27 \\ 4-1-20 & -2+1+12 & 3+4+36 \\ 20-3-45 & -10+2+27 & 15+12+81\end{array}\]\right]
=\left[\[\begin{array}{ccc}29 & -15 & -7 \\ -17 & 11 & 43 \\ -28 & 20 & 108\end{array}\]\right]
P^{2}=P^{2} \times P=\left[\[\begin{array}{ccc}29 & -15 & -7 \\ -17 & 11 & 43 \\ -28 & 20 & 108\end{array}\]\right]\left[\[\begin{array}{ccc}29 & -15 & -7 \\ -17 & 11 & 43 \\ -28 & 20 & 108\end{array}\]\right]
=\left[\[\begin{array}{lll}116-15-35 & -58+15+21 & 87+60+63 \\ -68+11+215 & 34-11-129 & -51-44-387 \\ -112+20+540 & 56-20-324 & -84-80-972\end{array}\]\right]
=\left[\[\begin{array}{ccc}66 & -22 & 210 \\ 158 & -106 & -482 \\ 448 & -288 & -1136\end{array}\]\right]
f(P)=\left[\[\begin{array}{ccc}66 & -22 & 210 \\ 158 & -106 & -482 \\ 448 & -288 & -1136\end{array}\]\right]-2\left[\[\begin{array}{ccc}29 & -15 & -7 \\ -17 & 11 & 43 \\ -28 & 20 & 108\end{array}\]\right]
+\left[\[\begin{array}{ccc}4 & -2 & 3 \\ 1 & -1 & -4 \\ 5 & -3 & -9\end{array}\]\right]-2\left[\[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{array}\]\right]
=\left[\[\begin{array}{ccc}66 & -22 & 210 \\ 158 & -106 & -482 \\ 448 & -288 & -1136\end{array}\]\right]-\left[\[\begin{array}{rcc}58 & -30 & -14 \\ -34 & 22 & 86 \\ -56 & 40 & 216\end{array}\]\right]+\left[\[\begin{array}{ccc}4 & -2 & 3 \\ 1 & -1 & -4 \\ 5 & -3 & -9\end{array}\]\right]-\left[\[\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\]\right]
=\left[\[\begin{array}{ccc}10 & 6 & 227 \\ 193 & -131 & -572 \\ 509 & -331 & -1363\end{array}\]\right]
\therefore F\left(A^{\top}\right)=\left[\[\begin{array}{ccc}10 & 6 & 227 \\ 193 & -131 & -572 \\ 509 & -331 & -1363\end{array}\]\right]
উত্তর
A=\left[\[\begin{array}{ccc}4 & 1 & 5 \\ -2 & -1 & -3 \\ 3 & -4 & -9\end{array}\]\right], X=\left[\[\begin{array}{l}x \y \z\end{array}\]\right], B=\left[\[\begin{array}{l}2 \\ 5 \\ 4\end{array}\]\right]
=\left[\[\begin{array}{ccc}4 & 1 & 5 \\ -2 & -1 & -3 \\ 3 & -4 & -9\end{array}\]\right]\left[\[\begin{array}{l}x \y \z\end{array}\]\right]=\left[\[\begin{array}{l}2 \\ 5 \\ 4\end{array}\]\right]
=\left[\[\begin{array}{ccc}4 x & 1 y & 5 z \\ -2 x & -1 y & -3 z \\ 3 x & -4 y & -9 z\end{array}\]\right]=\left[\[\begin{array}{l}2 \\ 5 \\ 4\end{array}\]\right]
এর সহগগুলো নিয়ে,
\mathrm{D}=\left[\[\begin{array}{ccc}4 & 1 & 5 \\ -2 & -1 & -3 \\ 3 & -4 & -9\end{array}\]\right]
D x=\left[\[\begin{array}{ccc}2 & 1 & 5 \\ 5 & -1 & -3 \\ 4 & -4 & -9\end{array}\]\right]
Dy =\left[\[\begin{array}{ccc}4 & 2 & 5 \\ -2 & 5 & -3 \\ 3 & 4 & -9\end{array}\]\right]
D z=\left[\[\begin{array}{ccc}4 & 1 & 2 \\ -2 & -1 & 5 \\ 3 & -4 & 4\end{array}\]\right]