Let $P = \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix}$ and $Q = q_{ij}$ be two $3 imes 3$ matrices such that $Q - P^{5} = I_{3}$. Then $\frac{q_{21} + q_{31}}{q_{32}}$ is equal to :

Question

Let $P = \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix}$ and $Q = q_{ij}$ be two $3 	imes 3$ matrices such that $Q - P^{5} = I_{3}$. Then $\frac{q_{21} + q_{31}}{q_{32}}$ is equal to :

Aicharya · Accepted Answer

Given, $P = \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix}$ $\Rightarrow P^{2} = \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 6 & 1 & 0 \ 27 & 6 & 1 \end{bmatrix}$ $\Rightarrow P^{4} = P^{2} 	imes P^{2} = \begin{bmatrix} 1 & 0 & 0 \ 6 & 1 & 0 \ 27 & 6 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \ 6 & 1 & 0 \ 27 & 6 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 12 & 1 & 0 \ 90 & 12 & 1 \end{bmatrix}$ $\Rightarrow P^{5} = P^{4} 	imes P = \begin{bmatrix} 1 & 0 & 0 \ 12 & 1 & 0 \ 90 & 12 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 9 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 15 & 1 & 0 \ 135 & 15 & 1 \end{bmatrix}$ Let, $Q = \begin{bmatrix} q_{11} & q_{12} & q_{13} \ q_{21} & q_{22} & q_{23} \ q_{31} & q_{32} & q_{33} \end{bmatrix}$ Now, given $Q - P^{5} = I_{3}$ $\Rightarrow q_{21} - 15 = 0, q_{31} - 135 = 0, q_{32} - 15 = 0$ $\Rightarrow q_{21} = 15, q_{31} = 135, q_{32} = 15$ $	herefore \frac{q_{21} + q_{31}}{q_{32}} = $\frac{15 + 135}{15}$ = 10$.

Let $P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$ and $Q = q_{ij}$ be two $3…

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