Question

$A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$ then $A^3-4A^2-6A$ is equal to :

• A. 0
• B. A
• C. -A
• D. I

Answer 1

Answer: (C)

$\therefore A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$
$\therefore A^2=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$
$=\left[\begin{array}{ccc}9& 8& 8\\ 8& 9& 8\\ 8& 8& 9\end{array}\right]$
$A^3=A^2\cdot A=\left[\begin{array}{ccc}9& 8& 8\\ 8& 9& 8\\ 8& 8& 9\end{array}\right]\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$
$=\left[\begin{array}{ccc}41& 42& 42\\ 42& 41& 42\\ 42& 42& 41\end{array}\right]$
$\therefore A^3-4A^2-6A$
$=\left[\begin{array}{ccc}41& 42& 42\\ 42& 41& 42\\ 42& 42& 41\end{array}\right]-4\left[\begin{array}{ccc}9& 8& 8\\ 8& 9& 8\\ 8& 8& 9\end{array}\right]-6\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$
$=\left[\begin{array}{ccc}41-36-6& 42-32-12& 42-32-12\\ 42-32-12& 41-36-6& 42-32-12\\ 42-32-12& 42-32-12& 41-36-6\end{array}\right]$
$=\left[\begin{array}{ccc}-1& -2& -2\\ -2& -1& -2\\ -2& -2& -1\end{array}\right]$
$=-\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]=-A$