If matrix A , = , [ * 20 c 1& 3k + 1 3 0&1 ], then the value of _… - Vidyadip

MCQ

If matrix $A\, = \,\left[ {\begin{array}{*{20}{c}}
1&{3k + \frac{1}{3}} \\
0&1
\end{array}} \right]$, then the value of $\mathop \Pi \limits_{k = 1}^{36} \,\left[ {\begin{array}{*{20}{c}}
1&{3k + \frac{1}{3}} \\
0&1
\end{array}} \right]$ is equal to :-

A
$\left[ {\begin{array}{*{20}{c}}
1&{1998} \\
0&1
\end{array}} \right]$
✓
$\left[ {\begin{array}{*{20}{c}}
1&{2010} \\
0&1
\end{array}} \right]$
C
$\left[ {\begin{array}{*{20}{c}}
1&{1005} \\
0&1
\end{array}} \right]$
D
$\left[ {\begin{array}{*{20}{c}}
1&{999} \\
0&1
\end{array}} \right]$

✓

Answer

Correct option: B.

$\left[ {\begin{array}{*{20}{c}}
1&{2010} \\
0&1
\end{array}} \right]$

b
$\prod\limits_{k = 1}^{36} {\left[ {\begin{array}{*{20}{c}}
1&{3k + \frac{1}{3}}\\
0&1
\end{array}} \right]} $

$ = \left[ {\begin{array}{*{20}{c}}
1&{3 + \frac{1}{3}}\\
0&1
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
1&{6 + \frac{1}{3}}\\
0&1
\end{array}} \right].......\left[ {\begin{array}{*{20}{c}}
1&{108 + \frac{1}{3}}\\
0&1
\end{array}} \right]$

$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
1&{\left( {3 + 6 + 9 + .... + 108} \right) + 12}\\
0&1
\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}
1&{2010}\\
0&1
\end{array}} \right]$

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