Question

Please do number 2

Assume all matricies are Mmxm(R) unless otherwise specified. 1. (1 point) Prove or disprove that the eigenvalues of A and AT are the same. 2. (2 points) Let A be a matrix with m distinct, non-zero, eigenvalues. Prove that the eigenvectors of A are linearly independent and span R”. Note that this means in this case) that the eigenvectors are distinct and form a base of the space. 3. (1 point) Given that is an eigenvalue of A associated with v, prove that v is an eigenvector of Ak and that lk is it's associated eigenvalue. 4. (2 points) Let A be a matrix with eigenvalues lk, 1 <k <m. Show that det A = 11 12 ... Im. 5. (1 point) Prove that a matrix A is singular if and only if it has a zero eigenvalue (hint: use the previous problem).

Answer 1

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