1. Let A be a square matrix with singular value decomposition A = UΣVT, prove that A is invertible if and only if all the singular values of A are nonzero.
2. Prove that the determinant of a square matrix is equal to the product of all its eigenvalues.
3. Use the result of the previous problem to prove that a square matrix is invertible if and only if its determinant is nonzero.
