Show that if a matrix A has zero as its eigenvalue it is not

Show that if a matrix A has zero as its eigenvalue, it is not invertible. Show that the linear transformation phi: R^N rightarrow R^H is one-to-one if its kernel is trivial (contains only zero) Suppose the linear transformation phi: R^N rightarrow R^H has the following property phi(phi(x)) = phi(x), for all x element of R^N Show that for all x element of R^H, x - phi(x) element of ker phi. Suppose A and B are two square matrices of the same size. Show that AB is invertible if and only if A and B are both invertible.

Solution

1. Assume 0 is an eigenvalue. Thus there is some nontrivial solution to Ax = 0x = 0. By the invertible matrix theorem, if A was invertible there would only be the trivial solution. Since there in a nontrivial solution, it must be the case that A is not invertible.

Hence proved if zero is an eigenvalue, the matrix is not invertible.