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Exponential Form through Prime Factorization

Let P = [ [...

Question

Let $P = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 \end{matrix} ⎤ ⎥ ⎦$ and $Q = [q_{i j}]$ be two $3 \times 3$ matrices such that $Q - P^{5} = I_{3} .$ Then $\frac{q_{21} + q_{31}}{q_{32}}$ is equal to:

A

15

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B

9

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C

135

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D

10

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Solution

The correct option is D $10$

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P = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 \end{matrix} ⎤ ⎥ ⎦

Q = [q_{i j}]

3 \times 3

matrices such that

Q - P^{5} = I_{3} .

\frac{q_{21} + q_{31}}{q_{32}}

P = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 \end{matrix} ⎤ ⎥ ⎦

Q = [q_{i j}]

3 \times 3

matrices such that

Q - P^{5} = I_{3} .

\frac{q_{21} + q_{31}}{q_{32}}

P = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 4 & 1 & 0 16 & 4 & 1 \end{matrix} ⎤ ⎥ ⎦

I

be the identity matrix of order

3

Q = [q_{i j}]

is a matrix, such that

p^{50} - Q = I

\frac{q_{31} + q_{32}}{q_{21}}

P = ⎡ ⎢ ⎣ \begin{matrix} 1416 \end{matrix} \begin{matrix} 014 \end{matrix} \begin{matrix} 001 \end{matrix} ⎤ ⎥ ⎦

I

be the identity matrix of order

3

Q = [q_{i j}]

is a matrix such that

P^{50} - Q = I

\frac{q_{31} + q_{32}}{q_{21}}

Q. Let P and Q be

3 \times 3

matrices such that

P \neq Q

P^{3} = Q^{3}

P^{2} Q = Q^{2} P

then determinant of

(P^{2} + Q^{2})

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