Check whether the matrix is square
Compute its determinant; if the determinant is nonzero, the matrix is invertible
Row-reduce the matrix; if it has a pivot in every row and every column, it is invertible
Verify that its rank equals its size
Check whether the only solution to Ax = 0 is the trivial solution x = 0
Determine whether its columns are linearly independent
Determine whether its rows are linearly independent
Check whether zero is an eigenvalue; if zero is not an eigenvalue, the matrix is invertible