031 Review Part 3, Problem 10

Show that if  ${\vec {x}}$  is an eigenvector of the matrix product  $AB$  and  $B{\vec {x}}\neq {\vec {0}},$  then  $B{\vec {x}}$  is an eigenvector of  $BA.$

Foundations:
An eigenvector  ${\vec {x}}$  of a matrix  $A$  is a nonzero vector such that
$A{\vec {x}}=\lambda {\vec {x}}$
for some scalar  $\lambda .$

Solution:

Step 1:

Step 2:

Final Answer:

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