Question

If $A=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right],$ where $A^x=O$ (where, $O$ is a null matrix and $x<15,x\in N$ ), then which of the following is true?

• A. Greatest value of $x$ is $13$
• B. Sum of the values of $x$ is $102$
• C. Difference between the largest and the smallest value of $x$ is $10$
• D. Number of values of $x$ is $7$

Answer 1

Answer: (B)

$A^2=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$ $\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$
$=\left[\begin{array}{ccc}0& 0& 0\\ 3& 3& 9\\ -1& -1& -3\end{array}\right]$
$A^3=A^{2}A=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]\left[\begin{array}{ccc}0& 0& 0\\ 3& 3& 9\\ -1& -1& -3\end{array}\right]$
$=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& -1& 0\end{array}\right]=O$
$A^4=A^5=A^6=\dots \dots A^{14}=O$
$x\in \left\{3,4,5,6,\dots \dots 14\right\}$