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Hi; The central problems of Linear Algebra are the solution of a simultaneous set of linear equations or Ax = b and determining the eigenvalues of a matrix. The eigenvalues are usually computed using a computer and we will not break with tradition, they are There is a little theorem that says if a square matrix has distinct eigenvalues then it is diagonalizable. So this one is diagonalizable. To do it we need the Eigenvectors of A: To check whether we have diagonalized it we plug in to Okay, so what? The useful fact is that to get A^k we only now need the following matrix equation. Now D^k is easy to get because to raise a matrix with just diagonal elements like D to the kth power you just take each element and raise it to the kth power. So if we wanted A^10 we would compute And we are done! 