Doing formally, you need calculate the determinant of the matrix as a polynomial of "t";

                    then find its roots (zeroes);

                    then analyze the matrix at every root value of t.



But, looking in this matrix, I see it MENTALLY, that one of the root is 1 (one, ONE)

    since at t= 1 all three lines of the matrix are identical, making the determinant equal to zero.


Also, due to the same reason, at t= 1  the rank of the matrix is 1.


It means, from the other side, that  t= 1  is the root of the determinant of the multiplicity 2.


Next, if you add the first and the second rows of the matrix to its third row,
you will get in the third row three
identical elements (t+2), (t+2), (t+2)


It means that t= -2 is the third root of the determinant of the multiplicity 1.


ANSWER.  The determinant is zero at t= 1  and  t= -2.

         The value of t= 1 is the zero of multiplicity 2;  the value of t= -2 is the zero of the multiplicity 1.


         The rank of the matrix is 1 at t= 1;  is 2 at t= -2  and is 3 at any other value of t.