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Question

Let P=[aij] be a 3×3 invertible matrix, where aij{0,1} for 1i,j3 and exactly four elements of P are 1. If N denotes the number of such possible matrices P, then which of the following is/are true?

A
Number of divisors of N is even.
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B
Sum of divisors of N is 91.
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C
Determinant of adj(P) can be 1.
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D
Determinant of adj(P) can be 1.
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Solution

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=1116 matrices for 4th 1

(ii) P=1116 matrices

(iii) P=1116 matrices

(iv) P=1116 matrices

(v) P=1116 matrices

(vi) P=1116 matrices

N=36=2232
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|31=(±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

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