Summary  
Computing the determinant of a square matrix via direct formulas (2×2 rule and cofactor expansion) and using library functions to assess invertibility and space‐scaling effects.

General domain of usage  
geometric transformations

**Video Description:**  
This video introduces determinants, demonstrates how to compute them for 2x2 and 3x3 matrices, and explores their geometric interpretation in terms of area, volume, and invertibility. You will see visualizations showing how the determinant measures scaling, and why a zero determinant means a matrix is not invertible.

The **determinant** is a special scalar value that you can compute from a square matrix. For a 2x2 matrix, the determinant can be found using a simple formula. If you have a matrix:

$$
A = \begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
$$

the determinant of $$A$$, often written as $$det(A)$$ or $$|A|$$, is calculated as:

$$
\text{det}(A) = ad - bc
$$

For a 3x3 matrix:

$$
B = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}
$$

the determinant is:

$$
\text{det}(B) = aei + bfg + cdh - ceg - bdi - afh
$$

**Geometrically**, for a 2x2 matrix, the absolute value of the determinant gives the **area** of the parallelogram formed by its column vectors. For a 3x3 matrix, the absolute value of the determinant gives the **volume** of the parallelepiped defined by its column vectors. If the determinant is zero, it means the area or volume collapses to zero, indicating that the matrix is **singular** and not invertible. A nonzero determinant means the matrix transformation preserves some volume or area and is **invertible**.

import numpy as np

# Example 2x2 matrix
A = np.array([[2, 3],
              [1, 4]])
det_A = np.linalg.det(A)
print("Determinant of A:", round(det_A))

# Example 3x3 matrix
B = np.array([[1, 2, 3],
              [0, 4, 5],
              [1, 0, 6]])
det_B = np.linalg.det(B)
print("Determinant of B:", round(det_B))

In the code above, you use `numpy.linalg.det()` to compute the determinant of each matrix directly. This built-in function efficiently calculates the determinant for any square matrix, so you do not need to apply the explicit formula yourself. 

The **determinant** is a key value for determining if a matrix is **invertible**. If the determinant is zero, the matrix is **singular** and cannot be inverted. If the determinant is nonzero, the matrix is **invertible**. This property is essential for solving linear systems and understanding how linear transformations affect the space.

Which of the following statements about determinants is correct?

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Determinants and Their Interpretation