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Let,A=\left[\begin{array}{lll} 0 & 1 & 2 \\ \mathrm{a} & 0 & 3 \\ 1 & \mathrm{c} & 0 \end{array}\right] , where a, c \epsilon \mathbb{R} . If A3 = A and the positive value of a belongs to the interval (n – 1, n],

where n\epsilon \mathbb{N} , then n is equal to ______:

Option: 1

2


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\begin{aligned} \mathrm{A} & =\left[\begin{array}{lll} 0 & 1 & 2 \\ \mathrm{a} & 0 & 3 \\ 1 & \mathrm{c} & 0 \end{array}\right] \\ \mathrm{A}^3 & =\mathrm{A} \\ \mathrm{A}^2 & =\left[\begin{array}{lll} 0 & 1 & 2 \\ \mathrm{a} & 0 & 3 \\ 1 & \mathrm{c} & 0 \end{array}\right]\left[\begin{array}{lll} 0 & 1 & 2 \\ \mathrm{a} & 0 & 3 \\ 1 & \mathrm{c} & 0 \end{array}\right] \\ \mathrm{A}^2 & =\left[\begin{array}{ccc} \mathrm{a}+2 & 2 \mathrm{c} & 3 \\ 3 & \mathrm{a}+3 \mathrm{c} & 2 \mathrm{a} \\ \mathrm{ac} & 1 & 2+3 \mathrm{c} \end{array}\right] \\ \mathrm{A}^3 & =\left[\begin{array}{ccc} \mathrm{a}+2 & 2 \mathrm{c} & 3 \\ 3 & \mathrm{a}+3 \mathrm{c} & 2 \mathrm{a} \\ \mathrm{ac} & \mathrm{a} & 2+3 \mathrm{c} \end{array}\right]\left[\begin{array}{lll} 0 & 1 & 2 \\ \mathrm{a} & 0 & 3 \\ 1 & \mathrm{c} & 0 \end{array}\right] \end{aligned}

\begin{aligned} & \mathrm{A}^3=\left[\begin{array}{ccc} 2 \mathrm{ac}+3 & \mathrm{a}+2+3 \mathrm{c} & 2 \mathrm{a}+4+6 \mathrm{c} \\ \mathrm{a}(\mathrm{a}+3 \mathrm{c})+2 \mathrm{a} & 3+2 \mathrm{ac} & 6+3 \mathrm{a}+9 \mathrm{c} \\ \mathrm{a}+2+3 \mathrm{c} & \mathrm{ac}+\mathrm{c}(2+3 \mathrm{c}) & 2 \mathrm{ac}+3 \end{array}\right] \\ & \text { Given } A^3=A \\ & 2 \mathrm{ac}+3=0 \ldots(1) \text { and } \mathrm{a}+2+3 \mathrm{c}=1 \\ & a+1+3 c=0 \\ & a+1-\frac{9}{2 a}=0 \\ & 2 a^2+2 a-9=0 \\ & \mathrm{f}(1)<0, \mathrm{f}(2)>0 \\ & \mathrm{a} \in(1,2] \\ & \mathrm{n}=2 \\ & \end{aligned}

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