Let A=[[|gathered 0 1 2 a 0 3 1 c 0 gathered]]|, where a,c R.

Question

Let $A=[[|\begin{gathered} 0 \quad  1 \quad  2 \ a \quad  0 \quad  3 \ 1 \quad  c \quad  0 \end{gathered}]]|$, where $a,c\in R$. If ${A}^{3}=A$ and the positive value of $a$ belongs to the interval $(n-1,n]$, where $n\in N$, then $n$ is equal to \_\_\_\_\_\_\_\_\_\_\_.

Answer: 2

Accepted Answer

$A=[[|\begin{gathered} 0 \quad  1 \quad  2 \ a \quad  0 \quad  3 \ 1 \quad  c \quad  0 \end{gathered}]]|a,c\in R$

$\begin{gathered} {A}^{2} \quad  =[[|\begin{array}{ccc}0 \quad  1 \quad  2 \ a \quad  0 \quad  3 \ 1 \quad  c \quad  0\end{array}]]|[[|\begin{array}{ccc}0 \quad  1 \quad  2 \ a \quad  0 \quad  3 \ 1 \quad  c \quad  0\end{array}]]| \ =[[|\begin{array}{ccc}a+2 \quad  2c \quad  3 \ 3 \quad  a+3c \quad  2a \ ac \quad  1 \quad  2+3c\end{array}]]| \end{gathered}$

$\begin{gathered} {A}^{3} \quad  =[[|\begin{array}{ccc}a+2 \quad  2c \quad  3 \ 3 \quad  a+3c \quad  2a \ ac \quad  1 \quad  2+3c\end{array}]]|[[|\begin{array}{ccc}0 \quad  1 \quad  2 \ a \quad  0 \quad  3 \ 1 \quad  c \quad  0\end{array}]]| \ =[[|\begin{array}{ccc}2ac+3 \quad  a+2+3c \quad  2a+4+6c \ a(a+3c)+2a \quad  3+2ac \quad  6+3a+9c \ a+2+3c \quad  ac+2c+3{c}^{2} \quad  2ac+3\end{array}]]| \end{gathered}$

$\begin{gathered} {A}^{3}=A[Given] \ 2ac+3=0a+2+3c=1 \ {a}^{2}+2a+3ac=a \ \Rightarrow {a}^{2}+a+3(-\frac{3}{2})=0 \ \Rightarrow 2{a}^{2}+2a-9=0 \end{gathered}$

When, $a=1,2{a}^{2}+2a-90$

$	herefore $ Positive value of $a\in (1,2]$

Hence, $n=2$

Solutions · 1