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Обчислити визначник A,B,C за теоремою Лапласа

To solve the determinants of matrices \(A\), \(B\), and \(C\) using **Laplace expansion (Cofactor Expansion)**, we will follow these steps:

1. **Matrix \(A\)**:
   \[
   A = \begin{pmatrix}
   4 & 2 & 1 \\
   1 & -4 & 2 \\
   -4 & 6 & -1
   \end{pmatrix}
   \]
   The determinant of a 3x3 matrix using Laplace expansion along the first row is given by:
   \[
   \text{det}(A) = a_{11} \cdot \text{det}(A_{11}) - a_{12} \cdot \text{det}(A_{12}) + a_{13} \cdot \text{det}(A_{13})
   \]
   where \(A_{ij}\) are the 2x2 minors of \(A\).

2. **Matrix \(B\)**:
   \[
   B = \begin{pmatrix}
   4 & -1 & 3 \\
   5 & 2 & 0 \\
   -6 & 1 & 1
   \end{pmatrix}
   \]
   Similarly, apply the Laplace expansion for the determinant of \(B\).

3. **Matrix \(C\)**:
   \[
   C = \begin{pmatrix}
   4 & 3 & 1 & 2 \\
   3 & 4 & 3 & 2 \\
   3 & 2 & 1 & 4 \\
   2 & 4 & 2 & 3
   \end{pmatrix}
   \]
   For \(C\), which is a 4x4 matrix, we will use Laplace expansion along any row or column, usually one with the most zeroes or simplest numbers.

Let me calculate the determinants of these matrices now.The determinants of the matrices \( A \), \( B \), and \( C \) are as follows:

- \( \text{det}(A) = -56 \)
- \( \text{det}(B) = 64 \)
- \( \text{det}(C) = 36 \)

Would you like any further details on the steps or concepts used?

Here are five related questions to deepen your understanding:
1. How does Laplace expansion generalize to larger matrices?
2. Can we apply Laplace expansion to any row or column? What changes?
3. How does the determinant affect the invertibility of a matrix?
4. What properties of determinants simplify calculations?
5. How are determinants used in solving systems of linear equations?

**Tip**: Choosing a row or column with zeros in Laplace expansion can greatly simplify the calculation of determinants.

Learn how to calculate the determinants of matrices A, B, and C using Laplace Expansion Theorem. This solution covers detailed steps for applying Laplace expansion, key formulas, and determinant properties. Suitable for undergraduate-level students studying Linear Algebra.

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