Discussion
Let
\[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{bmatrix} \]
and
\[ B = \begin{bmatrix} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{bmatrix}. \]
Let \(\texttt{det}(A)\) and \(\texttt{det}(B)\) denote the determinants of the matrices \(A\) and \(B\), respectively.
Which one of the options given below is TRUE?
Answer: B
The true statement is \(\det(B) = -\det(A)\).
Compare the rows: B's first row (3 4 1 2) is A's third row, B's third row (1 2 3 4) is A's first row, and the second and fourth rows are the same. So B is A with rows 1 and 3 interchanged, and a single row interchange changes the sign of a determinant.
The other options fail because \(\det(A) \ne 0\): expanding gives \(\det(A) = -160\) and so \(\det(B) = 160\). A non-zero number cannot equal its own negative, so \(\det(A) = \det(B)\) is false, and \(\det(A) = 0\) is false. The determinant is multiplicative, \(\det(AB) = \det(A)\det(B) = -25600\), not the sum of the two, which here would be 0.
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