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मान लीजिये \(\rm A = \begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&1\\\ 0&0&1&1\end{pmatrix}\) और ℝ4 पर सममित द्विरैखिक रूप (v, w) = vt Aw, जहाँ v, w ∈ ℝ4, पर विचार कीजिये। निम्नलिखित में से कौन सा कथन सत्य है?

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  1. A व्युत्क्रमणीय है
  2. ऐसे शून्येतर सदिश v, w विद्यमान हैं जिनके लिए 〈v, w〉 = 0
  3. सभी शून्येतर सदिशों u, v, w के लिए जहाँ v ≠ w, 〈u, v〉 ≠ 〈u, w〉
  4. A2 का प्रत्येक आइगेनमान धनात्मक है

Answer (Detailed Solution Below)

Option 2 : ऐसे शून्येतर सदिश v, w विद्यमान हैं जिनके लिए 〈v, w〉 = 0
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Detailed Solution

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सही उत्तर (2) है।

हम बाद में हल अपडेट करेंगे।
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Q1.For a quadratic form f(x, y, z) ∈ ℝ[x, y z], we say that (a, b, c) ∈ ℝ3 is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)? 
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