For each of the following, give an example if it exists. If it doesn't exist, explain why.
a) An invertible 3x3 matrix which is not diagonalizable.
c) An 3x3 matrix A with A^6+I3=0 (Hint: Use the determinant)
I know in order for a matrix to be diagonalizable, the matrix A has to be similar with D (Diagonal Matrix). Which means having the same eigenvalues... Since the eigenvalues are on the main diagonal. All invertible matrix cant have 0 in the main diagonal in the reduced echelon form so it has to be diagonalizable so it does not exist...I am pretty sure this is not a good explanation, so I am asking for some clarification on diagonalization properties for invertible matrixe.
for b): I used the determinant like this: DetA^6 = -detI3 => (DetA)^6= -1. So it does not exist...Is it what the question meant?
I am not good with algebra so forgive me for my misunderstandings.