Question

Let \[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{bmatrix} \] 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?

• A. \( \text{det}(A) = \text{det}(B) \)
• B. \( \text{det}(B) = -\text{det}(A) \)
• C. \( \text{det}(A) = 0 \)
• D. \( \text{det}(AB) = \text{det}(A) + \text{det}(B) \)

Hint:

When two matrices are related by a row or column swap, their determinants are negatives of each other.

Answer 1

The Correct Option is B

To solve this problem, we need to evaluate the determinants of matrices \( A \) and \( B \) and analyze their relationship. Let's proceed step by step.

1. Understanding Matrix Determinants

The determinant of a square matrix is a scalar value that is a function of its entries and provides important properties such as invertibility of the matrix.

2. Calculate \(\text{det}(A)\)

Let's evaluate the determinant of matrix \( A \): 

\( A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{bmatrix} \)

Using cofactor expansion or a direct calculator approach, we compute the determinant:

\(\text{det}(A) = -160\)

3. Calculate \(\text{det}(B)\)

Now let's calculate the determinant of matrix \( B \):

\( B = \begin{bmatrix} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{bmatrix} \)

Similarly, compute the determinant of \( B \):

\(\text{det}(B) = 160\)

4. Analyzing the Results

We observe that:

• \(\text{det}(A) = -160\)
• \(\text{det}(B) = 160\)

Hence, it follows that \(\text{det}(B) = -\text{det}(A)\).

5. Conclusion

The correct option is \(\text{det}(B) = -\text{det}(A)\).