Let P - - 1 0 0- 3 1 0- 9 3 1 - and Q - -qij- be two 3 - 3 matrices such that Q - P5 - I3- Then q21-q31q32 is equal to

P = nbsp;[ 1 amp; 0 amp;0; nbsp; 3 amp; 1 amp;0; nbsp; 9 amp;3 nbsp; amp; 1 ] ⇒ nbsp; P^2= nbsp;[ 1 amp; 0 amp;0; nbsp; 3 amp; ...

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Byju's Answer

Standard XIII

Mathematics

Multiplication of Matrices

Let P = [ 1 0...

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

9

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10

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15

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135

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Solution

The correct option is B $10$
$P = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow P^{2} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 6 & 1 & 0 27 & 6 & 1 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow P^{2} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 + 3 & 1 & 0 9 + 9 + 9 & 3 + 3 & 1 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow P^{3} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 + 3 + 3 & 1 & 0 (1 + 2 + 3) {.3}^{2} & 3 + 3 + 3 & 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ P^{n} = ⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 3 n & 1 & 0 \frac{n (n + 1)}{2} 3^{2} & 3 n & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦$
$\Rightarrow P^{5} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 15 & 1 & 0 135 & 15 & 1 \end{matrix} ⎤ ⎥ ⎦$
Now, $Q - P^{5} = I_{3}$
$Q = I_{3} + P^{5} = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & 0 15 & 2 & 0 135 & 15 & 2 \end{matrix} ⎤ ⎥ ⎦$
Hence, $\frac{q_{21} + q_{31}}{q_{32}} = \frac{15 + 135}{15} = 10$

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Similar questions

Q. 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:

Q. 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

Q. 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_{i j}]

is a matrix, such that

p^{50} - Q = I

, then

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

equals

Q. Let

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

and

I

be the identity matrix of order

3

. If

Q = [q_{i j}]

is a matrix such that

P^{50} - Q = I

, then

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

equals

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

and

I

be the identity matrix of order

3.

Q = [q_{i j}]

is a matrix such that

P^{50} - Q = I,

then

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

equals

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Let P=⎡⎢⎣100310931⎤⎥⎦ and Q=[qij] be two 3×3 matrices such that Q−P5=I3. Then q21+q31q32 is equal to

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