Consider the Matrix.   1. Diagonalise the matrix A, that is, find PDP^-1 2. Calculate the matrix power A^3 3. Calculate the determinant and the trace of D 4. Prove that for a diagonalisable square matrix A, the determinant is the product of the eigenvalues of A matrix and the trace is the sum of the eigenvalues of A. You may use that the trace is cyclic, that is, Tr(ABC) = Tr(CAB) = Tr(BCA), for any 3 square matrices A, B, C of the same size. 5. We saw earlier in the course that the derivative df(x) dx and the anti-derivative R f(x)dx of a single variable function are linear operators on the vector space of smooth functions (functions in one variable where derivatives and integrals are always well defined). What are the eigenvectors and eigenvalues of those operators?

Consider the Matrix.

1. Diagonalise the matrix A, that is, find PDP^-1

2. Calculate the matrix power A^3

3. Calculate the determinant and the trace of D

4. Prove that for a diagonalisable square matrix A, the determinant is the product of
the eigenvalues of A matrix and the trace is the sum of the eigenvalues of A. You
may use that the trace is cyclic, that is, Tr(ABC) = Tr(CAB) = Tr(BCA), for
any 3 square matrices A, B, C of the same size.

5. We saw earlier in the course that the derivative df(x)
dx and the anti-derivative R
f(x)dx of a single variable function are linear operators on the vector space of smooth
functions (functions in one variable where derivatives and integrals are always well
defined). What are the eigenvectors and eigenvalues of those operators?