# Probs in Linear Algebra

1 3

11. Consider (A) =

2 1

(a) Find the eigenvalue and corresponding eigenvectors of A.

(b) Determine matrices B and C such that B A C is diagonal.

(c) Show the eigenvectors of (A) are linearly independent.

(d) Represent the vector < 1,1> in terms of the eigenvectors of A.

12. Let, γi and γj be distinct eigenvalues of A with corresponding eigenvectors ei and ej show that ei and ej are linearly independent.

13. Consider the stochastic matrix

2/3 1/3

P=

1/4 3/4

(i) Find the eigenvalues of P.

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(ii) If A = , find A^-1

1 -3

Shown that (A^-1) (P) (A) = Diagonal and give the elements of it.

.

(iii) What are the eigenvectors of P?

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