Let X- 1- 1- 1 - and A- -1 2 3- 0 1 6- 0 0 -1- - For k- N- if X-AkX-33- then k is equal to

nbsp;X=[ 1; 1; 1 ] and A=[ 1 amp; 2 amp; 3; 0 amp; 1 amp; 6; 0 amp; 0 amp; 1; ] A^2=[ 1 amp; 0 amp; 6; 0 amp; 1 am ...

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Standard X

Mathematics

Equality of Matrices

Let X=[ 1; 1;...

Question

Let $X = ⎡ ⎢ ⎣ \begin{matrix} 111 \end{matrix} ⎤ ⎥ ⎦$ and $A = ⎡ ⎢ ⎣ \begin{matrix} - 1 & 2 & 3 0 & 1 & 6 0 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎦$ . For $k \in N$ , if $X^{'} A^{k} X = 33$ , then $k$ is equal to

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Solution

$X = ⎡ ⎢ ⎣ \begin{matrix} 111 \end{matrix} ⎤ ⎥ ⎦$ and $A = ⎡ ⎢ ⎣ \begin{matrix} - 1 & 2 & 3 0 & 1 & 6 0 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎦$

$A^{2} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 6 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$

$A^{3} = ⎡ ⎢ ⎣ \begin{matrix} - 1 & 2 & - 3 0 & 1 & 6 0 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎦$

$A^{4} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 12 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$

$A^{8} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 24 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$

Now, $A = ⎡ ⎢ ⎣ \begin{matrix} - 1 & 2 & 3 0 & 1 & 6 0 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} - 2 & 2 & 3 0 & 0 & 6 0 & 0 & - 2 \end{matrix} ⎤ ⎥ ⎦$

let $P = ⎡ ⎢ ⎣ \begin{matrix} - 2 & 2 & 3 0 & 0 & 6 0 & 0 & - 2 \end{matrix} ⎤ ⎥ ⎦$

$P^{2} = ⎡ ⎢ ⎣ \begin{matrix} 4 & - 6 & 6 0 & 0 & - 12 0 & 0 & 4 \end{matrix} ⎤ ⎥ ⎦$

$A^{5} = A^{4} \times P = ⎡ ⎢ ⎣ \begin{matrix} - 1 & 2 & - 9 0 & 1 & 6 0 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎦$

$A^{6} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 12 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$

$A^{10} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 24 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ \times ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 6 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 30 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$

$∴ K = 10$

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Similar questions

A sequence x_{1}, x_{2}, x_{3}, . . . is defined by letting x_{1} = 2 and x_{k} = \frac{x_{k - 1}}{k} for all natural numbers k, k \geq 2 . Show that x_{n} = \frac{2}{n!} for all n \in N .

[NCERT EXEMPLAR]

A sequence $x_{1}, x_{2}, x_{3}, . . . . . .$ is defined by letting $x_{1} = 2$ and $x_{k} = \frac{x_{k - 1}}{n}$ for all natural numbers k, $k \geq 2$ . Show that $x_{n} = \frac{2}{n!}$ for all $n ϵ N$ .

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{lim}_{n \to \infty} \frac{(n + 1)^{k - 1}}{n^{k + 1}} [(n k + 1) + (n k + 2) + \dots + (n k + n)] = 33 \cdot {lim}_{n \to \infty} \frac{1}{n^{k + 1}} \cdot [1^{k} + 2^{k} + 3^{k} + \dots + n^{k}]

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Let X=⎡⎢⎣111⎤⎥⎦ and A=⎡⎢⎣−12301600−1⎤⎥⎦. For k∈N, if X′AkX=33, then k is equal to

Let $X = ⎡ ⎢ ⎣ \begin{matrix} 111 \end{matrix} ⎤ ⎥ ⎦$ and $A = ⎡ ⎢ ⎣ \begin{matrix} - 1 & 2 & 3 0 & 1 & 6 0 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎦$ . For $k \in N$ , if $X^{'} A^{k} X = 33$ , then $k$ is equal to