All matrices have a pseudoinverse. However, I cannot prove this for a singular square matrix.
The pseudoinverse of a matrix is given as
$$A^{+} = (A^{T}A)^{-1}A^{T}$$
$$(A^{T}A)^{-1} = \frac{C^{T}}{det(A^{T}A)} = \frac{C^{T}}{det(A^{T})det(A)}$$
So wouldn't
$$(A^{T}A)^{-1}$$ be singular, which means the a pseudoinverse does not exist for a singular square matrix. Am I wrong? If so how?