Question

A matrix M_r is defined as \(M_r = \begin{bmatrix} r & r - 1 \\\ r - 1 & r \end{bmatrix} r \in N\), then the value of det (M₁) + det(M₂) + ... + det(M₂₀₁₅) is
1. 2014²
2. 2013²
3. 2015
4. 2015²

Answer 1

Correct Answer - Option 4 : 2015²

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