Question

If the matrix $A=\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$ , then the value of $\frac{\left|A^{100} + A^{98}\right|}{\left|A^{20} + A^{18}\right|}$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (B)

$\left|A\right|=\begin{vmatrix} 2 & 5 \\ 1 & 3 \end{vmatrix}=6-5=1$
$\frac{\left|A^{100} + A^{98}\right|}{\left|A^{20} + A^{18}\right|}=\frac{\left|A^{98}\right| \left|A^{2} + I\right|}{\left|A^{18}\right| \left|A^{2} + I\right|}=\left|A\right|^{80}=1$