The determinants of the reverse identity matrices are defined as follow:∣J1∣=∣1∣=+1,∣J2∣=∣0110∣=−1,∣J3∣= \left|J_{1} ight|=|1|=+1,\left|J_{2} ight|=\left|\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array} ight|=-1,\left|J_{3} ight|= ∣001010100∣=−1,⋯ \left|\begin{array}{lll}0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0\end{array} ight|=-1, \cdots Find the determinant of J100 \mathrm{J}_{100} .

Question

The determinants of the reverse identity matrices are defined as follow:∣J1∣=∣1∣=+1,∣J2∣=∣0110∣=−1,∣J3∣= \left|J_{1}ight|=|1|=+1,\left|J_{2}ight|=\left|\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array}ight|=-1,\left|J_{3}ight|= ∣001010100∣=−1,⋯ \left|\begin{array}{lll}0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0\end{array}ight|=-1, \cdots Find the determinant of J100 \mathrm{J}_{100} .

Accepted Answer

ক. +1

ব্যাখ্যা: $n 	imes n$ আকারের বিপরীত অভেদক ম্যাট্রিক্স (reverse identity matrix) $J_n$-এর নির্ণায়কের মান $|J_n| = (-1)^{n(n-1)/2}$।
এখানে $n = 100$ হলে,
$\frac{n(n-1)}{2} = \frac{100 	imes 99}{2} = 50 	imes 99 = 4950$ (যা একটি জোড় সংখ্যা)।
অতএব, $|J_{100}| = (-1)^{4950} = +1$।

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