Question

Matrix $M_r$ is defined as $M_r=\left[\begin{array}{cc}r& r-1\\ r-1& r\end{array}\right],rϵN$ value of $\text{det}\left(M_1\right)+\text{det}\left(M_2\right)+\text{det}\left(M_3\right)+....+\text{det}\left(M_{2007}\right)$ is

• A. $2007$
• B. $2008$
• C. $(2008{)}^2$
• D. $(2007{)}^2$

Answer 1

Answer: (D)

$(2007{)}^2$

$\text{det}\left(M_r\right)=\left|\begin{array}{cc}r& r-1\\ r-1& r\end{array}\right|=2r-1$
$\sum _{r=1}^{2007}\text{det}\left(M_r\right)=2\sum _{r=1}^{2007}(r-2007)$
$=2\times \frac{2007\times 2008}{2}-2007$
$=(2007{)}^2$.