JEE MAIN - Mathematics (2021 - 25th July Evening Shift - No. 14)
If $$P = \left[ {\matrix{
1 & 0 \cr
{{1 \over 2}} & 1 \cr
} } \right]$$, then P50 is :
$$\left[ {\matrix{
1 & 0 \cr
{25} & 1 \cr
} } \right]$$
$$\left[ {\matrix{
1 & {50} \cr
0 & 1 \cr
} } \right]$$
$$\left[ {\matrix{
1 & {25} \cr
0 & 1 \cr
} } \right]$$
$$\left[ {\matrix{
1 & 0 \cr
{50} & 1 \cr
} } \right]$$
Explanation
$$P = \left[ {\matrix{
1 & 0 \cr
{{1 \over 2}} & 1 \cr
} } \right]$$
$${P^2} = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right]\left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right] = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]$$
$${P^3} = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]\left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right] = \left[ {\matrix{ 1 & 0 \cr {{3 \over 2}} & 1 \cr } } \right]$$
$${P^4} = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]\left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right] = \left[ {\matrix{ 1 & 0 \cr 2 & 1 \cr } } \right]$$
$$ \vdots $$
$$\therefore$$ $${P^{50}} = \left[ {\matrix{ 1 & 0 \cr {25} & 1 \cr } } \right]$$
$${P^2} = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right]\left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right] = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]$$
$${P^3} = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]\left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right] = \left[ {\matrix{ 1 & 0 \cr {{3 \over 2}} & 1 \cr } } \right]$$
$${P^4} = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]\left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right] = \left[ {\matrix{ 1 & 0 \cr 2 & 1 \cr } } \right]$$
$$ \vdots $$
$$\therefore$$ $${P^{50}} = \left[ {\matrix{ 1 & 0 \cr {25} & 1 \cr } } \right]$$
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