Question

Define $A=\left[\begin{array}{cc}0& 1\\ 3& 0\end{array}\right]$. Find a vertical vector V such that $(A^8+A^6+A^4+A^2+I)V=\left[\begin{array}{c}0\\ 11\end{array}\right]$
(where I is the $2\times 2$ identity matrix).

• A. $V=\left[\begin{array}{c}0\\ \frac{-1}{9}\end{array}\right]$
• B. $V=\left[\begin{array}{c}0\\ \frac{1}{9}\end{array}\right]$
• C. $V=\left[\begin{array}{c}0\\ \frac{-1}{11}\end{array}\right]$
• D. $V=\left[\begin{array}{c}0\\ \frac{1}{11}\end{array}\right]$

Answer 1

Answer: (D)

$V=\left[\begin{array}{c}0\\ \frac{1}{11}\end{array}\right]$

$A=\left[\begin{array}{cc}0& 1\\ 3& 0\end{array}\right]$

$A^2=\left[\begin{array}{cc}0& 1\\ 3& 0\end{array}\right]\times \left[\begin{array}{cc}0& 1\\ 3& 0\end{array}\right]=\left[\begin{array}{cc}0+3& 0+0\\ 0+0& 3+0\end{array}\right]=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$

$A^4=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]\times \left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]=\left[\begin{array}{cc}9+0& 0+0\\ 0+0& 9+0\end{array}\right]=\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]$

$A^6=A^4\times A^2=\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]\times \left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]=\left[\begin{array}{cc}27& 0\\ 0& 27\end{array}\right]$

$A^8=\left[\begin{array}{cc}81& 0\\ 0& 81\end{array}\right]$

Now $A^8+A^6+A^4+A^2+I$

$=\left[\begin{array}{cc}81& 0\\ 0& 81\end{array}\right]+\left[\begin{array}{cc}27& 0\\ 0& 27\end{array}\right]+\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]+\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$=\left[\begin{array}{cc}121& 0\\ 0& 121\end{array}\right]$

$\therefore (A^8+A^6+A^4+A^2+I)V=\left[\begin{array}{c}0\\ 11\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}121& 0\\ 0& 121\end{array}\right]V=\left[\begin{array}{c}0\\ 11\end{array}\right]\Rightarrow V=\frac{1}{121\times 121}\left[\begin{array}{cc}121& 0\\ 0& 121\end{array}\right]\times \left[\begin{array}{c}0\\ 11\end{array}\right]$

$\Rightarrow V=\left[\begin{array}{c}0\\ \frac{1}{11}\end{array}\right]$