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    Probs in Linear Algebra

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    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, &#947;i and &#947;j 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.

    1 4
    (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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