... 16. Suppose that A is a 6 × 6 matrix with characteristic polynomial cA (λ) = (1 + λ)(1 − λ)2 (2 − λ)3 .
(a) Prove that it is impossible to find three linearly independent vectors vi , i = 1, 2, 3, such
that Avi = vi , i = 1, 2, 3.
... Let v denote a column vector of the nilpotent matrix
Pi (A)(A − λi I)ni −1
where ni is the so called nilpotency. Theorem 3 in  shows that
APi (A)(A − λi I)ni −1 = λi Pi (A)(A − λi I)ni −1 .
which means a column vector v of the matrix is an eigenvector corresponding
to the eigenvalue λi . The symb ...
... • When businesses deal with sales, there is a need to organize
information. For instance, let's say Karadimos King is a fast-food
restaurant that made the following number of sales:
• On Monday, Karadimos King sold 35 hamburgers, 50 sodas, and 45
fries. On Tuesday, it sold 120 sodas, 56 fries, and 4 ...
... 12. True or false: AT A = AAT for every n × n matrix A. Justify your answer.
13. True or false: Every subspace U ⊂ Rn is the null space (same as kernel) of a linear transformation T : Rn → Rk for some k. Justify your answer.
14. An n × n matrix is said to be symmetric if AT = A and anti-symmetric if ...