This is a practice exam for an exam that I\'m having on Monday. I want to make s

Question

This is a practice exam for an exam that I'm having on Monday. I want to make sure I have the right answers and the right way to get them. So, I would be grateful if you could solve these problems in detail. 1) Given the triangular matrix T: T = 1 2 3 0 2 1 0 0 3 A) Compute its exponent e^(xT) B) Compute the inverse of the exponent. ------------------------------- 2)Given the matrix M: M = 4 1 2 1 4 2 1 3 3 A) Find its eigenvalues. B) Find its eigenvectors. C) Set up the matrix P with the three eigenvectors and compute its inverse P^(-1) D) compute P^(-1)MP ------------------------------- 3) Given the matrix M: M = 4 2 2 2 4 2 2 2 4 A) Find its three orthogonal eigenvectors. B) Set up the matrix P with the three eigenvectors and compute the transpose of P.

Explanation / Answer

2) M = 4 1 2 1 4 2 1 3 3 A) eigen values => trace (M ) = 4+4+3 = 11 det(M) = 21 hence det(M - Lambda*I) = 0 where Lambda = a1 a2 a3 hence characteristic eqn is : x^3 - 11x^2 + 19X +27 = 0 a1 = 7 a2 = 1 a3 = 3 to find eigen vectors : Mx lambda*X 4x1 + x2 2X3 = 7X1 x1 + x2 + 2X3 = 7X2 X1 +3X2 + 3X3 = 7X3 we get one eigen vector as 1 1 1 similarly other eigen vectors are ( 1 1 -2 )

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