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Question 13 : 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

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From the information given in the question, each row must contain two 1’s and one -1. Similarly, each column must contain two 1’s and one -1. Because only 1+1+(-1) = 1 So the matrix can be \\left[\begin{array}{ccc}-1 & 1 & 1\\1 & -1 & 1\\1 & 1 & -1\end{array}\right]\\) or \\left[\begin{array}{ccc}-1 & 1 & 1\\1 & 1 & -1\\1 & -1 & 1\end{array}\right]\\) or \\left[\begin{array}{ccc}1 & -1 & 1\\-1 & 1 & 1\\1 & 1 & -1\end{array}\right]\\) or \\left[\begin{array}{ccc}1 & -1 & 1\\1 & 1 & -1\\-1 & 1 & 1\end{array}\right]\\) or \\left[\begin{array}{ccc}1 & 1 & -1\\-1 & 1 & 1\\1 & -1 & 1\end{array}\right]\\) or \\left[\begin{array}{ccc}1 & 1 & -1\\1 & -1 & 1\\-1 & 1 & 1\end{array}\right]\\)

Let us find the determinant of the first matrix and call it Det.

Det = -1((-1*-1)-(1*1)) -1((1*-1)-(1*1))+1((1*1)-(1*-1)) = -1(0)-1(-2)+1(2) = 4

We can convert the remaining 5 matrices into the first matrix by performing row operations,

If we perform even number of row operations, then the determinant of that matrix will be 4 itself.

If we perform odd number of row operations, then the determinant of that matrix will be -4.

So, the value of the determinant could be 4 or -4 but the absolute value of the determinant will always be |-4| or |4| which is equal to 4.

The absolute value of the determinant is 4.

The question is **"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" **

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