IPMAT Question Paper 2019 | IPM Indore Quants

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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

Explanatory Answer

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"

Hence, the answer is 4

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