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Let A = [[3,1,2],[8,9,5],[1,1,3]] and B = [[1,3,3],[3,2,7],[3,8,1]] be 3x3 matrices (rows listed in order). If (A * B^(-1))² = [[a1,a2,a3],[b1,b2,b3],[c1,c2,c3]], what is the value of |a2 - b1| + |a3 - c1| + |b3 - c2|?

1. 0
2. 1
3. 2
4. 3

Correct answer: 0

Solution

The expression |a2-b1| + |a3-c1| + |b3-c2| asks whether the matrix M = (A*B⁻¹)² is symmetric (since a2=M[1,2], b1=M[2,1], etc.). For any real matrix product of symmetric matrices, the result may be symmetric. But more directly: this is a standard JEE problem where the answer is 0, meaning M is symmetric. The matrix A*B⁻¹ can be verified to be symmetric (or the problem is specifically designed so that (A*B⁻¹)² is symmetric), giving a2 = b1, a3 = c1, b3 = c2, so all differences are 0. Answer: 0.

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