Question
Let \[P = \left[ {\begin{array}{*{20}{c}}
1&0&0\\
4&1&0\\
{16}&4&1
\end{array}} \right]\] and $$I$$ be the identity matrix of order 3. If $$Q = \left[ {{q_{ij}}} \right]$$ is a matrix such that $${P^{50}} - Q = I,{\text{then }}\frac{{{q_{31}} + {q_{32}}}}{{{q_{21}}}}$$ equals
A.
52
B.
103
C.
201
D.
205
Answer :
103
Solution :
\[\begin{array}{l}
P = \left[ \begin{array}{l}
1\,\,\,\,\,\,\,0\,\,\,\,\,\,0\\
4\,\,\,\,\,\,\,1\,\,\,\,\,\,0\\
16\,\,\,\,\,4\,\,\,\,\,1
\end{array} \right] = I + \left[ \begin{array}{l}
0\,\,\,\,\,\,\,0\,\,\,\,\,\,\,0\\
4\,\,\,\,\,\,\,0\,\,\,\,\,\,\,0\\
16\,\,\,\,\,4\,\,\,\,\,\,\,0
\end{array} \right] = I + A\\
{A^2} = \left[ \begin{array}{l}
\,0\,\,\,\,\,\,0\,\,\,\,\,\,0\\
\,0\,\,\,\,\,\,0\,\,\,\,\,\,0\\
16\,\,\,\,\,0\,\,\,\,\,\,0
\end{array} \right]\,\,{\rm{and }}\,\,{A^3} = \left[ \begin{array}{l}
0\,\,\,\,\,\,\,0\,\,\,\,\,\,0\\
0\,\,\,\,\,\,\,0\,\,\,\,\,\,0\\
0\,\,\,\,\,\,\,0\,\,\,\,\,\,0
\end{array} \right]
\end{array}\]
$$\eqalign{
& \therefore \,\,{A^n} = O,\forall \,\,n \geqslant 3 \cr
& {\text{Now }}{P^{50}} = {\left( {I + A} \right)^{50}} = {\,^{50}}{C_0}\,{I^{50}} + {\,^{50}}{C_1}\,{I^{49}}A + {\,^{50}}{C_2}\,{I^{48}}{A^2} + O \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = I + 50A + 25 \times 49{A^2}. \cr
& \therefore \,\,Q = {P^{50}} - I = 50A + 25 \times 49{A^2}. \cr
& \Rightarrow \,\,{q_{21}} = 50 \times 4 = 200 \cr
& \Rightarrow \,\,{q_{31}} = 50 \times 16 + 25 \times 49 \times 16 = 20400 \cr
& \Rightarrow \,\,{q_{32}} = 50 \times 4 = 200 \cr
& \therefore \,\,\frac{{{q_{31}} + {q_{32}}}}{{{q_{21}}}} = \frac{{20600}}{{200}} = 103 \cr} $$