Explanation: A এর কর্নের উপাদানগুলোর গুণফল = \(a^3\)
\(\therefore a^3 = 2\sqrt{2} = (\sqrt{2})^3 \therefore a = \sqrt{2}\)
এখন \(\sqrt{2}I - A = \sqrt{2}\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} - \begin{bmatrix} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)
\(\therefore \left| (\sqrt{2}I - A)^3 \right| = \left| \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \right| = 0\)
Another Explanation (5): ```html
সমাধান:
দেওয়া আছে, \( 3 \times 3 \) আকারের কর্ণ ম্যাট্রিক্স \( A \) এর কর্ণ উপাদানগুলোর গুণফল \( 2\sqrt{2} \)। ধরি, \( A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \)।
তাহলে, \( abc = 2\sqrt{2} \).
এখন, \( \sqrt{2}I - A = \begin{bmatrix} \sqrt{2}-a & 0 & 0 \\ 0 & \sqrt{2}-b & 0 \\ 0 & 0 & \sqrt{2}-c \end{bmatrix} \).
সুতরাং, \( |\sqrt{2}I - A| = (\sqrt{2}-a)(\sqrt{2}-b)(\sqrt{2}-c) \).
আমরা বের করতে চাই \( |(\sqrt{2}I - A)^3| \) এর মান।
আমরা জানি, \( |A^n| = |A|^n \).
সুতরাং, \( |(\sqrt{2}I - A)^3| = |\sqrt{2}I - A|^3 = [(\sqrt{2}-a)(\sqrt{2}-b)(\sqrt{2}-c)]^3 \).
এখন, \( (\sqrt{2}-a)(\sqrt{2}-b)(\sqrt{2}-c) = (\sqrt{2})^3 - \sqrt{2}^2(a+b+c) + \sqrt{2}(ab+bc+ca) - abc \)
\( = 2\sqrt{2} - 2(a+b+c) + \sqrt{2}(ab+bc+ca) - 2\sqrt{2} \)
\( = - 2(a+b+c) + \sqrt{2}(ab+bc+ca) \)
কিন্তু আমাদের \( |(\sqrt{2}I - A)^3| \) এর মান বের করতে হবে যখন \( abc = 2\sqrt{2} \).
এখন, \( |\sqrt{2}I - A|^3 = [(\sqrt{2}-a)(\sqrt{2}-b)(\sqrt{2}-c)]^3 \)
\( = [2\sqrt{2} - 2(a+b+c) + \sqrt{2}(ab+bc+ca) - abc]^3 \)
\( = [2\sqrt{2} - 2(a+b+c) + \sqrt{2}(ab+bc+ca) - 2\sqrt{2}]^3 \)
\( = [- 2(a+b+c) + \sqrt{2}(ab+bc+ca)]^3 \)
যদি \( a = b = c = \sqrt{2} \) হয়, তবে \( abc = (\sqrt{2})^3 = 2\sqrt{2} \) এবং
\( (\sqrt{2}I - A) = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)
সুতরাং, \( |(\sqrt{2}I - A)^3| = |0|^3 = 0 \).
অন্যথায়, ধরি \( a = 2, b = 1, c = \frac{\sqrt{2}}{2} \). সেক্ষেত্রে, \( abc = 2 \cdot 1 \cdot \frac{\sqrt{2}}{2} = \sqrt{2} \neq 2\sqrt{2} \)
যদি \( a = 2\sqrt{2}, b=1, c=1 \) হয়, তবে \( abc = 2\sqrt{2} \).
তখন \( |\sqrt{2}I - A| = (\sqrt{2} - 2\sqrt{2})(\sqrt{2}-1)(\sqrt{2}-1) = -\sqrt{2}(2 - 2\sqrt{2} + 1) = -\sqrt{2}(3 - 2\sqrt{2}) = -3\sqrt{2} + 4 \)
সুতরাং, \( |(\sqrt{2}I - A)^3| = (-3\sqrt{2} + 4)^3 \neq 0 \)
যদি কর্ণ উপাদানগুলো \(\sqrt{2}\) হয়, তবে \( A = \sqrt{2}I \). সেক্ষেত্রে \(\sqrt{2}I - A = 0\). সুতরাং, \( |(\sqrt{2}I - A)^3| = |0| = 0 \). 🥳
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