r is defined as \(M r = r & r - 1..." - Step-by-step solution, answer key, and detailed explanation on MCA Prep."> r is defined as \(M r = r & r - 1..." - Step-by-step solution, answer key, and detailed explanation on MCA Prep." /> r is defined as \(M r = r & r - 1..." - Step-by-step solution, answer key, and detailed explanation on MCA Prep." />
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Question 34

Mathematics Determinants Hard

A matrix M<sub>r</sub>&nbsp;is defined as&nbsp;<span class="math-tex">\(M_r = \begin{bmatrix} r &amp; r - 1 \\\ r - 1 &amp; r \end{bmatrix} r \in N\)</span>, then the value of det (M<sub>1</sub>) + det(M<sub>2</sub>) + ... + det(M<sub>2015</sub>) is

(A) 2014<sup>2</sup>
(B) 2013<sup>2</sup>
(C) 2015
(D) 2015<sup>2</sup>
✓ Correct Answer: Option D Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

\(\det(M_r) = r^2 - (r-1)^2 = 2r - 1\)

Sum = \(\sum_{r=1}^{2015}(2r - 1)\). The sum of the first \(n\) odd numbers is \(n^2\), so this is \(2015^2\).

Answer: (D) \(2015^2\)

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