$A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to 1. find the possible value of the determinant Let $A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to $1$. And remaining nine entries of the row being $0$. Which of the following is not a possible value of the determinant?
$0, 1 ,-1, 10$.
I am able to identify for $2\times 2$ cross two matrices for which possible value of determinant is $1$ or $-1$. How to identify for such a big size matrix? Can we identify such matrix?
 A: Adding or subtracting a row of a matrix from another does not change its determinant, so we may assume each column of the matrix has at most one entry that is 1.
Swapping rows of a matrix changes the sign of the determinant only; so if we perform row swaps so that the resulting matrix is diagonal, we'll have determined the determinant up to a sign.
So now we have a diagonal matrix whose diagonal entries are either 1 or 0.
The determinant of this matrix must be $0$ or $1$; and hence, the determinant of the original matrix must be $0$, $1$, or $-1$. 
(The $-1$ possibility can arise: start with the identity matrix and interchange the last two rows. The 0 possibility can arise: start with a matrix whose first column is all $1$'s. And, of course, the identity matrix shows that $1$ is a possible value of the determinant.)
A: The possible values of Determinant will be {0,1,-1}. Let me explain how.
We can consider A to be either an identity matrix(All the diagonal elements are 1). Hence it will satisfy the condition for A(exactly one entry in each row equal to 1). The determinant value of the Identity matrix is 1.
Now in this identity matrix, if we interchange a row/column, the sign of the value of determinant changes. So it becomes -1.
Now, since there is no condition that a column also has to have a single 1 throughout the column, hence we can consider that there are two identical rows in A, such that the position of 1 is the same in both the rows. Since, if a determinant contains identical rows/columns, its value is 0, therefore 0 is also a possible value for A.
