Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 24/Solution 1

Following the hint, the diagonal matrix $D$ is the eigenvalues and the column of $V$ are the are the corresponding right eigenvectors.

Therefore, A has 3 eigenvalues: $1+2i$, $1-2i$, $-1$, and their corresponding eigenvectors are $[0.8165,0,0.4082-0.4082i{]}^T$ , $[0.8165,0,0.4082+0.4082i{]}^T$ , $[0.5774,0.5774,0.5774{]}^T$. The eigenvectors in $V$ are normalized so that the 2-norm of each one is $1$.

So $(a)$ is wrong. We have $-1$

$(b)$ is wrong. We have $1+2i$

$(c),(d)$ is correct. A has three different eigenvalues and hence the three eigenvectors are independent, and they can form a set of basis.

$(e)$ is correct. $[1,1,1{]}^T$ is a scalar multiples of the last column vectors of $A$. The scalar multiplication of any eigenvector is a eigenvector as well, so $(e)$ is correct

Thus, the answer is $(c),(d),(e)$.