Question

$IfthematrixA=\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]then{A}^8-5A^7+7A^6-3A^5+A^4-5A^3+8A^2-2A+Iwillbe$______

• A. $\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]$
• B. $\left[\begin{array}{ccc}5& 4& 4\\ 0& 1& 0\\ 4& 4& 5\end{array}\right]$
• C. $\left[\begin{array}{ccc}7& 5& 5\\ 0& 2& 0\\ 5& 5& 7\end{array}\right]$
• D. $\left[\begin{array}{ccc}8& 5& 5\\ 0& 3& 0\\ 5& 5& 8\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{ccc}8& 5& 5\\ 0& 3& 0\\ 5& 5& 8\end{array}\right]$

The characteristic equation of the matrix A is
$\left[\begin{array}{ccc}(2-\lambda )& 1& 1\\ 0& (1-\lambda )& 0\\ 1& 1& (2-\lambda )\end{array}\right]=0$

${\lambda }^3-5{\lambda }^2+7\lambda -3=0$

According to cayley -Hamilton theorem, we have
$A^3-5A^2+7A-3I=0$
$A^5(A^3-5A^2+7A-3I)+A(A^3-5A^2+7A-3I)+A^2+A+I$
$=A^2+A+I$

$A^2=\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]=\left[\begin{array}{ccc}5& 4& 4\\ 0& 1& 0\\ 4& 4& 5\end{array}\right]$

$A^2+A+I=\left[\begin{array}{ccc}5& 4& 4\\ 0& 1& 0\\ 4& 4& 5\end{array}\right]+\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}8& 5& 5\\ 0& 3& 0\\ 5& 5& 8\end{array}\right]$