Correct Answer - Option 4 : 2015
2
Concept:
\(|M_r| = \begin{vmatrix} r & r - 1 \\\ r - 1 & r \end{vmatrix} r \in N\)\(\rm |M_r| = 2r - 1\)
\(\rm \sum _{r=1}^n n = \dfrac {n(n+1)}{2}\)
Calculations:
A matrix Mr is defined as \(M_r = \begin{bmatrix} r & r - 1 \\\ r - 1 & r \end{bmatrix} r \in N\)
⇒\(|M_r| = \begin{vmatrix} r & r - 1 \\\ r - 1 & r \end{vmatrix} r \in N\)
⇒\(\rm |M_r| = 2r - 1\)
Now, consider det (M1) + det(M2) + ... + det(M2015)
= \(\rm \sum_{r=1}^{2015} |M_r|= 2\sum_{r=1}^{2015} r - 2015\)
= \(\rm \sum_{r=1}^{2015} |M_r|= 2\times\dfrac { 2015\times 2016}{2} - 2015\)
= 20152