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  4. Matrix M r is defined as Mr= beginpmatrixr&r-1 r-1&r endpmatrix, r ∈ N The value det (M1) + det (M2)+ det (m3) + dots + det (M2014) is

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Q. Matrix $M _{ r }$ is defined as $M_{r}= \begin{pmatrix}r&r-1\\ r-1&r\end{pmatrix}, r \in N $ The value det $(M_{1})$ + det $(M_{2})$+ det $(m_{3})$ $+\dots +$ det $(M_{2014})$ is

Determinants

Solution:

det $(Mr) = \begin{vmatrix}r&r-1\\ r-1&r\end{vmatrix}=2r-1$
$\displaystyle\sum_{ r =1}^{2014} ( M _{ r })=2 \displaystyle\sum_{ r =1}^{2014} r -2014$
$=2 \times \frac{2014 \times 2015}{2}-2014=(2014)^{2}$