Let P = and I be the identity matrix of order 3. If Q = [qij] is a matrix such that P50

Engineering

Mathematics

Multiplication of Matrices

Question

Let P = $[\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}]$ and I be the identity matrix of order 3. If Q = [q_ij] is a matrix such that P⁵⁰ – Q = I, then $\frac{q_{31} + q_{32}}{q_{21}}$ equals

205

201

103

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P²=^{$[\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 8 & 1 & 0 \\ 48 & 8 & 1 \end{matrix}]$}

P³ = $[\begin{matrix} 1 & 0 & 0 \\ 8 & 1 & 0 \\ 48 & 8 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 12 & 1 & 0 \\ 96 & 12 & 1 \end{matrix}]$

P⁵⁰ = $[\begin{matrix} 1 & 0 & 0 \\ 200 & 1 & 0 \\ 16 (1 + 2 + 3 + \dots + 50) & 200 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 200 & 1 & 0 \\ 20400 & 200 & 1 \end{matrix}]$

Q = P⁵⁰ – I = $[\begin{matrix} 0 & 0 & 0 \\ 200 & 0 & 0 \\ 20400 & 200 & 0 \end{matrix}]$

Now, $\frac{q_{31} + q_{32}}{q_{21}}$ = 103. Ans.