Let A - 1 1 3- 5 2 6- -2 -1 -3 - Find the smallest n- N- such that An-O-

Consider, #160; A ^ 2 =[ 1 amp; 1 amp; 3; 5 amp; 2 amp; 6; 2 amp; 1 amp; 3 ][ 1 amp; 1 amp; 3; 5 amp; 2 amp; 6; ...

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Byju's Answer

Standard XII

Mathematics

Global Maxima

Let A =[ 1 ...

Question

Let $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 3 5 & 2 & 6 - 2 & - 1 & - 3 \end{matrix} ⎤ ⎥ ⎦ .$ Find the smallest $n \in N,$ such that $A^{n} = O$ .

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Solution

Consider, $A^{2} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 3 5 & 2 & 6 - 2 & - 1 & - 3 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 3 5 & 2 & 6 - 2 & - 1 & - 3 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow A^{2} = ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 3 & 3 & 9 - 1 & - 1 & - 3 \end{matrix} ⎤ ⎥ ⎦$
Now, $A^{3} = ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 3 & 3 & 9 - 1 & - 1 & - 3 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 3 5 & 2 & 6 - 2 & - 1 & - 3 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow A^{3} = ⎡ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦$
Hence, $A^{3} = O$
So, $n = 3$

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Let A=⎡⎢⎣113526−2−1−3⎤⎥⎦. Find the smallest n∈N, such that An=O.

Consider, A2=⎡⎢⎣113526−2−1−3⎤⎥⎦⎡⎢⎣113526−2−1−3⎤⎥⎦⇒A2=⎡⎢⎣000339−1−1−3⎤⎥⎦Now, A3=⎡⎢⎣000339−1−1−3⎤⎥⎦⎡⎢⎣113526−2−1−3⎤⎥⎦⇒A3=⎡⎢⎣000000000⎤⎥⎦Hence, A3=OSo, n=3

Let $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 3 5 & 2 & 6 - 2 & - 1 & - 3 \end{matrix} ⎤ ⎥ ⎦ .$ Find the smallest $n \in N,$ such that $A^{n} = O$ .