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

Question

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 (A) 2013 (B) 2014 (C) (2013)2 (D) (2014)2

Tardigrade Admin · Accepted Answer

det (Mr) = |r&r-1 r-1&r|=2r-1 displaystyle∑ r =12014 ( M r )=2 displaystyle∑ r =12014 r -2014 =2 × (2014 × 2015/2)-2014=(2014)2

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

2013

2014

$(2013)^{2}$

$(2014)^{2}$

det $(M r) = ∣ ∣ r r - 1 r - 1 r ∣ ∣ = 2 r - 1$
$r = 1 \sum 2014 (M_{r}) = 2 r = 1 \sum 2014 r - 2014$
$= 2 \times \frac{2014 \times 2015}{2} - 2014 = (2014)^{2}$

Q. Matrix Mr​ is defined as Mr​=(rr−1​r−1r​),r∈N The value det (M1​) + det (M2​)+ det (m3​) +⋯+ det (M2014​) is

2013

2014

(2013)2

(2014)2

det (Mr)=∣∣​rr−1​r−1r​∣∣​=2r−1 r=1∑2014​(Mr​)=2r=1∑2014​r−2014 =2×22014×2015​−2014=(2014)2

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

$(2013)^{2}$

$(2014)^{2}$

det $(M r) = ∣ ∣ r r - 1 r - 1 r ∣ ∣ = 2 r - 1$
$r = 1 \sum 2014 (M_{r}) = 2 r = 1 \sum 2014 r - 2014$
$= 2 \times \frac{2014 \times 2015}{2} - 2014 = (2014)^{2}$