Question

If $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]$ and I is the unit matrix of order 3, then $A^2+2A^4+4A^6$ is equal to

• A. $7A^8$
• B. $7A^7$
• C. $8I$
• D. $6I$

Answer 1

Answer: (A)

$A^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$A^2=A^4=A^6=I_3\Rightarrow A^2+2A^4+4A^6$
$=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]+\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]$
$=\left[\begin{array}{ccc}7& 0& 0\\ 0& 7& 0\\ 0& 0& 7\end{array}\right]=7I_3=7A^8$