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Diagonizable Matrix, Inverse and Nullspace

1) If a Matrix A is diagonizable, must it have an inverse ? if so, is it diagonizable? Can {see attachment} be diagonized, does it have an inverse as well as {see attachment}

2) A is mxn
For m<n, is there a vector b such that Ax = b does not have any solution? Any trivial solution for Ax = 0?
b) Can say the same for m>n ? Any possible dimension for nullspace of A?


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Diagonizable Matrices, Inverses and Nullspaces are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.