If a 3 X 3 matrix is filled with +1 's and - 1 's such that the sum of each row and column of the matrix is 1, then the absolute value of its determinant is

The only way by which we can add 1 and -1 to get 1 is using two 1's and one -1.

or, we have to use (1,1,-1) in any order to fill the 3 X 3 matrix

Also, it is mentioned that  sum of each row and column of the matrix has to be 1

This means the order in which we fill $$i^{th}$$ row, in the same order have to fill $$i^{th}$$ coloumn

or, we can say elements of matrix $$a_{ij}=a_{ji}$$

or, the matrix has to be symmetric matrix.

The following examples are possible:

$$\begin{bmatrix}1&-1 & 1 \\ -1 &1&1\\ 1 & 1 & -1 \end{bmatrix}$$ , $$\begin{bmatrix}1&1 & -1 \\ 1 &-1&1\\ -1 & 1 & 1 \end{bmatrix}$$ etc.

Let's calculate absolute value of determinant for any one of them.

Let's take matrix $$\begin{bmatrix}1&1 & -1 \\ 1 &-1&1\\ -1 & 1 & 1 \end{bmatrix}$$

It's determinant will be $$\begin{vmatrix}1&1 & -1 \\ 1 &-1&1\\ -1 & 1 & 1 \end{vmatrix}$$

Applying $$R_1->R_1+R_2$$,

$$\begin{vmatrix}1&1 & -1 \\ 1 &-1&1\\ -1 & 1 & 1 \end{vmatrix}$$-->$$\begin{vmatrix}2&0 & 0 \\ 1 &-1&1\\ -1 & 1 & 1 \end{vmatrix}$$

Now breaking the determinant along $$R_1$$,

Value=2$$\begin{vmatrix}-1 & 1 \\ 1 & 1 \end{vmatrix}$$=2(-1-1)=-4

So, absolute value will be 4