The correct option is D Determinant of adj(P) can be 1.
We know that, determinant of an invertible matrix is non-zero.
Since, each elements of matrix P is 0 or 1. Hence, there must be atleast a 1 in each row and each column. So, possible cases are
(i) P=⎡⎢⎣1−−−1−−−1⎤⎥⎦≡6 matrices for 4th 1
(ii) P=⎡⎢⎣1−−−−1−1−⎤⎥⎦≡6 matrices
(iii) P=⎡⎢⎣−1−1−−−−1⎤⎥⎦≡6 matrices
(iv) P=⎡⎢⎣−1−−−11−−⎤⎥⎦≡6 matrices
(v) P=⎡⎢⎣−−1−1−1−−⎤⎥⎦≡6 matrices
(vi) P=⎡⎢⎣−−11−−−1−⎤⎥⎦≡6 matrices
∴N=36=22⋅32
Divisors of N are
1,2,3,4,6,9,12,18,36.
Number of divisors =(2+1)(2+1)=9
Sum of divisiors =91
|adj(P)|=|P|3−1=(±1)2=1
Alternate method :
Finding N by using permutation and combination
Choosing one row, so the possible ways =3C1
Choosing one column, so the possible ways =2C1
Third one is fixed.
Now, there are 6 places for the fourth 1, so
N=3×2×6=36