The matrix is


    P = 



If you write all 6 the determinant's terms, 4 of them (out-the-diagonals-terms) will contain 0 (zero) as a factor, 

and, therefore, will be equal to zero.



The only two diagonal terms will contribute to the determinant


    det(P) = (t+2)*(t-1)*(3t+1) - (2t+4)*(t-1)*(t+1) = (t-1)*((t+2)*(3t+1) - (2t+4)*(t+1)) = 

           = (t-1)*(t+2)*((3t+1) - 2*(t+1)) = (t-1)*(t+2)*(t-1) = (t+2)*(t-1)^2.



The zeroes of the determinant are the values t= -2 (of multiplicity 1) and t= 1 (of multiplicity 2).