\b[False.] Take the <diagonal matrix> $A=\operatorname{diag}(1,2)$ and its <eigenvectors> $\mathbf x=(1,0)^T$, $\mathbf y=(0,1)^T$. Their sum $(1,1)^T$ is mapped to $(1,2)^T$, which is not a <scalar multiple> of that sum. More generally, $A(\mathbf x+\mathbf y)=a\mathbf x+c\mathbf y$ need not have the form $\lambda(\mathbf x+\mathbf y)$ when $a\ne c$.

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 1 / Section B / 8 / a / i / Solution

 Solution (source code)