Science:Math Exam Resources/Courses/MATH152/April 2022/Question A15/Solution 1

Since we want ${\textstyle A}$ to have 2 linearly independent eigenvectors associated with ${\textstyle \lambda =1}$, then we would have ${\textstyle \mathbf {rank} (A-\lambda I)=1}$. This means that using row operations, we should be able to reduce the matrix ${\textstyle A-\lambda I}$ to a single row! We need to find ${\textstyle b}$ such that this holds.

$$A-\lambda I=\left[{\begin{array}{ccc}0&-4&b\\0&2&3\\0&0&0\end{array}}\right]\rightarrow \left[{\begin{array}{ccc}0&2&3\\0&0&b+6\\0&0&0\end{array}}\right].$$

Then, in order to have ${\textstyle A-\lambda I}$ reduce to a single row, we take ${\textstyle b=-6}$.