JEE MAIN - Mathematics (2023 - 11th April Morning Shift - No. 17)
Let $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$$, where $$a, c \in \mathbb{R}$$. If $$A^{3}=A$$ and the positive value of $$a$$ belongs to the interval $$(n-1, n]$$, where $$n \in \mathbb{N}$$, then $$n$$ is equal to ___________.
Answer
2
Explanation
$$
\text { We have, } A=\left[\begin{array}{lll}
0 & 1 & 2 \\
a & 0 & 3 \\
1 & c & 0
\end{array}\right] \text {, where } a, c \in R
$$
$$ \begin{aligned} A^2 & =\left[\begin{array}{lll} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{array}\right]\left[\begin{array}{lll} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{array}\right] \\\\ & =\left[\begin{array}{ccc} a+2 & 2 c & 3 \\ 3 & a+3 c & 2 a \\ a c & 1 & 2+3 c \end{array}\right] \end{aligned} $$
$$ \begin{aligned} A^3 & =\left[\begin{array}{ccc} a+2 & 2 c & 3 \\ 3 & a+3 c & 2 a \\ a c & 1 & 2+3 c \end{array}\right]\left[\begin{array}{ccc} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{array}\right] \\\\ & =\left[\begin{array}{ccc} 2 a c+3 & a+2+3 c & 2 a+4+6 c \\ a(a+3 c)+2 a & 3+2 a c & 6+3 a+9 c \\ a+2+3 c & a c+2 c+3 c^2 & 2 a c+3 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} & A^3 =A [Given]\\\\ & 2 a c+3= 0 \text { and } a+2+3 c=1 \\\\ & a^2+2 a+3 a c =a \\\\ & \Rightarrow a^2 +a+3\left(-\frac{3}{2}\right)=0\\\\ & \Rightarrow 2 a^2+2 a-9=0 \end{aligned} $$
When, $a=1,2 a^2+2 a-9<0$ and
When, $a=2,2 a^2+2 a-9>0$
$\therefore$ Positive value of $a \in(1,2]$
Hence, $n=2$
$$ \begin{aligned} A^2 & =\left[\begin{array}{lll} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{array}\right]\left[\begin{array}{lll} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{array}\right] \\\\ & =\left[\begin{array}{ccc} a+2 & 2 c & 3 \\ 3 & a+3 c & 2 a \\ a c & 1 & 2+3 c \end{array}\right] \end{aligned} $$
$$ \begin{aligned} A^3 & =\left[\begin{array}{ccc} a+2 & 2 c & 3 \\ 3 & a+3 c & 2 a \\ a c & 1 & 2+3 c \end{array}\right]\left[\begin{array}{ccc} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{array}\right] \\\\ & =\left[\begin{array}{ccc} 2 a c+3 & a+2+3 c & 2 a+4+6 c \\ a(a+3 c)+2 a & 3+2 a c & 6+3 a+9 c \\ a+2+3 c & a c+2 c+3 c^2 & 2 a c+3 \end{array}\right] \end{aligned} $$
$$ \begin{aligned} & A^3 =A [Given]\\\\ & 2 a c+3= 0 \text { and } a+2+3 c=1 \\\\ & a^2+2 a+3 a c =a \\\\ & \Rightarrow a^2 +a+3\left(-\frac{3}{2}\right)=0\\\\ & \Rightarrow 2 a^2+2 a-9=0 \end{aligned} $$
When, $a=1,2 a^2+2 a-9<0$ and
When, $a=2,2 a^2+2 a-9>0$
$\therefore$ Positive value of $a \in(1,2]$
Hence, $n=2$
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