Question

If $A=\left(\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right)$
If $A^2-4A=pI$ where $I$ and $O$ are the unit matrix and the null matrix of order $3$ respectively. Find the value of $p$

• A. $p=2$
• B. $p=3$
• C. $p=4$
• D. $p=5$

Answer 1

The correct option is D $p=5$

Given
$A=\left(\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right)$
$\therefore A^2=A.A=\left(\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right)\times \left(\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right)$
$=\left(\begin{array}{ccc}1+4+4& 2+2+4& 2+4+2\\ 2+2+4& 4+1+4& 4+2+2\\ 2+4+2& 4+2+2& 4+4+1\end{array}\right)$
$=\left(\begin{array}{ccc}9& 8& 8\\ 8& 9& 8\\ 8& 8& 9\end{array}\right)$
$\therefore A^2-4A=\left(\begin{array}{ccc}9& 8& 8\\ 8& 9& 8\\ 8& 8& 9\end{array}\right)-4\left(\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right)$
$=\left(\begin{array}{ccc}9& 8& 8\\ 8& 9& 8\\ 8& 8& 9\end{array}\right)-\left(\begin{array}{ccc}4& 8& 8\\ 8& 4& 4\\ 4& 4& 8\end{array}\right)=\left(\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 5\end{array}\right)=5I$