{\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]\left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]$$

$$ = \left[ {\matrix{ 1 & 0 & 0 \cr {4 + 4} & 1 & 0 \cr {16 + 32} & {4 + 4} & 1 \cr } } \right]$$

$$ = \left[ {\matrix{ 1 & 0 & 0 \cr {4 \times 2} & 1 & 0 \cr {16(1 + 2)} & {4 \times 2} & 1 \cr } } \right]$$ ...... (i)

and $${P^3} = \left[ {\matrix{ 1 & 0 & 0 \cr {4 \times 2} & 1 & 0 \cr {16(1 + 2)} & {4 \times 2} & 1 \cr } } \right]\left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]$$

$$ = \left[ {\matrix{ 1 & 0 & 0 \cr {4 \times 3} & 1 & 0 \cr {16(1 + 2 + 3)} & {4 \times 3} & 1 \cr } } \right]$$ ..... (ii)

From symmetry,

$${P^{50}} = \left[ {\matrix{ 1 & 0 & 0 \cr {4 \times 50} & 1 & 0 \cr {16(1 + 2 + 3 + .... + 50)} & {4 \times 50} & 1 \cr } } \right]$$

$$\because$$ $${P^{50}} - Q = I$$ [given]

$$\therefore$$ $$\left[ {\matrix{ {1 - {q_{11}}} & { - {q_{12}}} & { - {q_{13}}} \cr {200 - {q_{21}}} & {1 - {q_{22}}} & { - {q_{23}}} \cr {16 \times {{50} \over 2}(51) - {q_{31}}} & {200 - {q_{32}}} & {1 - {q_{33}}} \cr } } \right]$$

$$ = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$

$$ \Rightarrow 200 - {q_{21}} = 0$$, $${{16 \times 50 \times 51} \over 2} - {q_{31}} = 0$$, $$200 - {q_{32}} = 0$$

$$\therefore$$ $${q_{21}} = 200$$, $${q_{32}} = 200$$, $${q_{31}} = 20400$$

Thus, $${{{q_{31}} + {q_{32}}} \over {{q_{21}}}} = {{20400 + 200} \over {200}}$$

$$ = {{20600} \over {200}} = 103$$

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