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Question: If the matrix M<sub>r</sub> is given by M<sub>r</sub> = \(\begin{bmatrix} r & r - 1 \\ r - 1 & r \en...

If the matrix Mr is given by Mr = [rr−1r−1r]\begin{bmatrix} r & r - 1 \\ r - 1 & r \end{bmatrix}, r = 1, 2, 3 .... Then the value of det (M1) + det(M2) + .... + det (M2009) is

A

2008

B

2009

C

(2009)2

D

(2008)2

Answer

(2009)2

Explanation

Solution

Mr = [rr−1r−1r]\begin{bmatrix} r & r - 1 \\ r - 1 & r \end{bmatrix}

|Mr| = r2 – (r – 1)2 = 2r – 1

Ž ∑r=1n∣Mr∣\sum_{r = 1}^{n}{|M_{r}|} = ∑r=1n(2r−1)\sum_{r = 1}^{n}{(2r - 1)} = n2