Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 15/Hint 3

Eigenvectors are very specific to a matrix. Multiplying a matrix A with e₁ returns the first column of A. That is, if a_j is the j-th column of A,

$A={\begin{bmatrix}a_{1}&|&a_{2}&|&\dots &|&a_{n}\end{bmatrix}}$

then

$Ae_{1}={\begin{bmatrix}a_{1}&|&a_{2}&|&\dots &|&a_{n}\end{bmatrix}}e_{1}=a_{1}$

Since v = e₁ is an eigenvector of the matrix A we know that

${\begin{aligned}Av&=\lambda v\\Ae_{1}&=\lambda e_{1}={\begin{bmatrix}\lambda \\0\\\vdots \\0\end{bmatrix}}\end{aligned}}$

Hence a₁ = λ e₁. In other words, the only non-zero entry of the first column of A is in the first row. Can you perform elementary row operations to move that entry into another row, and hence find a counter example?