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Let A = [1 2 3 4 4 1 2 3 3 4 1 2 2 3 4 1] and B = [3 4 1 2 4 1 2 3 1 2 3 4 2 3 4 1]. Let det(A) and det(B) denote the determinants of the matrices A and B, respectively. Which one of the options given below is TRUE?

1. det(A) = det(B)
2. det(B) = -det(A)
3. det(A) = 0
4. det(AB) = det(A) + det(B)

Correct answer: det(B) = -det(A)

Solution

The correct option is true because matrix B can be obtained from matrix A by a series of row operations that include swapping rows, which changes the sign of the determinant. Therefore, the relationship between their determinants is that det(B) equals the negative of det(A).

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