Q45 / 160 · AP EAMCET 2022 (05 Jul Shift 1) · Mathematics

If $A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$, then $\operatorname{det}\left(A^6+B^6\right)=$

  • A
    $-68$
  • Correct answer
    B
    $-212$
  • C
    $665$
  • D
    $720$

Solution

Given, $A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$ $\begin{aligned} & \Rightarrow A^2=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right] \\ & \Rightarrow A^2=\left[\begin{array}{ll}1+0 & 0+0 \\ 2+2 & 0+1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 4 & 1\end{array}\right]\end{aligned}$ $\Rightarrow A^3=\left[\begin{array}{ll}1 & 0 \\ 4 & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right] \quad\left[A^2 \cdot A=A^3\right]$ $\Rightarrow A^3 \doteq\left[\begin{array}{cc}1+0 & 0+0 \\ 4+2 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 6 & 1\end{array}\right]$ Similarly, $A^6=\left[\begin{array}{cc}1 & 0 \\ 12 & 1\end{array}\right] \quad\left[A^n=A_{21}=n \times 2\right]$ Now, $B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$ $\begin{aligned} & \Rightarrow B^2=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right] \\ & \Rightarrow B^2=\left[\begin{array}{ll}1+0 & 3+3 \\ 0+0 & 0+1\end{array}\right]=\left[\begin{array}{ll}1 & 6 \\ 0 & 1\end{array}\right] \\ & \Rightarrow B^3=\left[\begin{array}{ll}1 & 6 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 9 \\ 0 & 1\end{array}\right]\end{aligned}$ Similarly, $B^6=\left[\begin{array}{cc}1 & 18 \\ 0 & 1\end{array}\right]$ $\left[B^n=B_{12}=n \times 3\right]$ $\therefore \quad\left(A^6+B^6\right)=\left[\begin{array}{cc}1 & 0 \\ 12 & 1\end{array}\right]+\left[\begin{array}{cc}1 & 18 \\ 0 & 1\end{array}\right]$ $A^6+B^6=\left[\begin{array}{cc}2 & 18 \\ 12 & 2\end{array}\right]$ $\operatorname{Det}\left(A^6+B^6\right)=4-216=-212$

Hence, the correct answer is option: 'B'.

Relevant Questions