https://r2.erweima.ai/mathbot/image/1726087801054x9tdj1a7.jpg

The image contains a problem about determinants. Here's a breakdown of the information provided in the problem:

1. The matrix \( A \) is given as:
   \[
   A = \begin{pmatrix}
   a & b & c \\
   d & e & f \\
   1 & 2 & 3
   \end{pmatrix}
   \]
   and its determinant \( |A| = 2 \).

2. Another matrix \( B \) is provided:
   \[
   B = \begin{pmatrix}
   6c & 2b & 2a \\
   3f & e & d \\
   9 & 2 & 1
   \end{pmatrix}
   \]
   and the determinant of this matrix is \( |B| \).

3. The relation between the determinants \( |A| \) and \( |B| \) is used to find \( |C| \), where
   \[
   |C| = \left| \frac{1}{2} A^{-1} A^{t} \right|
   \]
   which is equal to \( \frac{1}{8} \) in the selected answer.

### Solution Strategy:

- The problem states \( |A| = 2 \).
- Matrix \( B \) is a transformation of matrix \( A \), and we need to find the determinant \( |B| \). Through the matrix multiplication and determinant properties, \( |B| = -12 \), as selected in option c).
- \( |C| \) is based on the properties of the inverse and transpose of matrix \( A \), leading to the result \( |C| = \frac{1}{8} \), consistent with the option c).

Thus, the correct answer is indeed option **c) \( |B| = -12 \) and \( |C| = \frac{1}{8} \)**.



Sabiendo que |A| = 2, los determinantes |B| y |C| deben ser hallados usando las propiedades de matrices dadas.

This problem explores the determinants of matrices, where |A| is given as 2, and the goal is to find |B| and |C| using matrix transformations. Key concepts include the inverse and transpose of matrices. The correct solution yields |B| = -12 and |C| = 1/8.

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