If the matrix Mr is given by Mr=[ beginmatrix r r-1 r-1 r endmatrix ],r=1,2,3,...., then the value of det (M1)+ det (M2)+....+ det (M2008) is

Question

If the matrix Mr is given by Mr=[ beginmatrix r r-1 r-1 r endmatrix ],r=1,2,3,...., then the value of det (M1)+ det (M2)+....+ det (M2008) is (A) 2007 (B) 2008 (C)  (2008)2  (D)  (2007)2

Tardigrade Admin · Accepted Answer

Given, Mr=[ beginmatrix r r-1 r-1 r endmatrix ] ∴ det (Mr)=r2-(r-1)2=2r-1 ∴ det (M1)+ det (M2)+....+ det (M2008) =1+3+5+...+4015 =(2008/2)[2+(2008-1)2] =2008(2008)=(2008)2

Q. If the matrix $M_{r}$ is given by $M_{r} = [r r - 1 r - 1 r], r = 1, 2, 3, ....,$ then the value of $det (M_{1}) + det (M_{2}) + .... + det (M_{2008})$ is

$2007$

$2008$

$(2008)^{2}$

$(2007)^{2}$

$2009$

Given, $M_{r} = [r r - 1 r - 1 r]$
$∴$ $det (M_{r}) = r^{2} - (r - 1)^{2} = 2 r - 1$
$∴$ $det (M_{1}) + det (M_{2}) + .... + det (M_{2008})$
$= 1 + 3 + 5 + ... + 4015$
$= \frac{2008}{2} [2 + (2008 - 1) 2]$
$= 2008 (2008) = (2008)^{2}$

Q. If the matrix Mr​ is given by Mr​=[rr−1​r−1r​],r=1,2,3,...., then the value of det(M1​)+det(M2​)+....+det(M2008​) is

2007

2008

(2008)2

(2007)2

2009

Given, Mr​=[rr−1​r−1r​] ∴ det(Mr​)=r2−(r−1)2=2r−1 ∴ det(M1​)+det(M2​)+....+det(M2008​) =1+3+5+...+4015 =22008​[2+(2008−1)2] =2008(2008)=(2008)2