Science:Math Exam Resources/Courses/MATH152/April 2022/Question A07/Solution 1

If ${\vec {v}}$ is an eigenvector of $A$, then we have $$\left[{\begin{array}{cc}1&-1\\-3&a\end{array}}\right]\left[{\begin{array}{c}1\\3\end{array}}\right]=\lambda \left[{\begin{array}{c}1\\3\end{array}}\right],$$ for some real number $\lambda$. In other words, we have the system of equations

$${\begin{cases}1-3&=\lambda \\-3+3a&=3\lambda .\end{cases}}$$

Solving the above system shows that $a=-1$.