Question

If $A=\left[\begin{array}{cc}1& 4\\ 2& 3\end{array}\right]and{A}^5-4A^4-7A^3+11A^2-A-10I=A+xI,$ then x is

• A. 2
• B. 3
• C. 5
• D. none

Answer 1

The correct option is C 5

Given, $A=\left[\begin{array}{cc}1& 4\\ 2& 3\end{array}\right]$

Characteristic equation of A is,

$\left|\begin{array}{cc}1-\lambda & 4\\ 2& 3-\lambda \end{array}\right|=0$

$(1-\lambda )(3-\lambda )-8=0$
${\lambda }^2-4\lambda -5=0$
Now,

${\lambda }^5-4{\lambda }^4-7{\lambda }^3+11{\lambda }^2-\lambda -10$

$=({\lambda }^2-4\lambda -5)({\lambda }^3-2\lambda +3)+(\lambda +5)$
$=\lambda +5$

$\therefore A^5-4A^4-7A^3+11A^2-A-10I=A+5I$
$\therefore x=5$