Question

If $A=\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix},$ 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)

$A^{2}=\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}$ $\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}$
$=\begin{bmatrix} 0 & 0 & 0 \\ 3 & 3 & 9 \\ -1 & -1 & -3 \end{bmatrix}$
$A^{3}=A^{2}A=\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}\begin{bmatrix} 0 & 0 & 0 \\ 3 & 3 & 9 \\ -1 & -1 & -3 \end{bmatrix}$
$=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & -1 & 0 \end{bmatrix}=O$
$A^{4}=A^{5}=A^{6}=\ldots \ldots A^{14}=O$
$x\in \left\{3 , 4 , 5 , 6 , \ldots \ldots 14\right\}$