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 $\det(A)$ and $\det(B)$ denote the determinants of the matrices $A$ and $B$, respectively. Which one of the options given below is TRUE?

Let the rows of matrix $A$ be denoted by $R_1, R_2, R_3, R_4$:

• $R_1 = \begin{bmatrix} 1 & 2 & 3 & 4 \end{bmatrix}$

• $R_2 = \begin{bmatrix} 4 & 1 & 2 & 3 \end{bmatrix}$

• $R_3 = \begin{bmatrix} 3 & 4 & 1 & 2 \end{bmatrix}$

• $R_4 = \begin{bmatrix} 2 & 3 & 4 & 1 \end{bmatrix}$

Comparing with matrix $B$:

• Row 1 of $B$ is $R_3$

• Row 2 of $B$ is $R_2$

• Row 3 of $B$ is $R_1$

• Row 4 of $B$ is $R_4$

Matrix $B$ is obtained from matrix $A$ by swapping Row 1 and Row 3 ($R_1 \leftrightarrow R_3$).

Interchanging any two rows of a matrix multiplies its determinant by $-1$. Therefore:

Matrix $A$ is a circulant matrix with non-zero determinant (specifically, $\det(A) = -160 \neq 0$), so $\det(A) \neq \det(B)$ and $\det(A) \neq 0$. Thus, option B is TRUE.