I am requested to search out the attribute polynomial of the matrix: $A = smallbegin{pmatrix}
2 & 2 & 0 & 0
2 & 2 & 0 & 0
0 & 0 & 2 & 2
0 & 0 & 0 & 4
finish{pmatrix}$. I’ve gotten $$start{equation}
start{cut up}
p_A(t) &= det (A-tI)
&= det smallbegin{pmatrix}
2-t & 2 & 0 & 0
2 & 2-t & 0 & 0
0 & 0 & 2-t & 2
0 & 0 & 0 & 4-t
finish{pmatrix}
finish{cut up}
finish{equation}$$

Nevertheless, the error I appear to make comes from my analysis of $det smallbegin{pmatrix}
2-t & 2 & 0 & 0
2 & 2-t & 0 & 0
0 & 0 & 2-t & 2
0 & 0 & 0 & 4-t
finish{pmatrix}$.

My approach of evaluating is:
$$start{equation} start{cut up}
det smallbegin{pmatrix}
2-t & 2 & 0 & 0
2 & 2-t & 0 & 0
0 & 0 & 2-t & 2
0 & 0 & 0 & 4-t
finish{pmatrix} &= (2-t)(2-t)(2-t)(4-t) – 0
&=(2-t)^3(4-t).
finish{cut up} finish{equation}$$

After I use an eigenvalue calculator to verify my polynomial, I see $p_A(t) = (4-t)^2(2-t)t$. The step-by-step answer says to search out the determinant of $A-tI$ it’s a must to cut back $A-tI$ to REF. My query is: Why do we have now to cut back $A-tI$ to its REF?