Determinant

💡 What is the Determinant?

For a square matrix $A \in \mathbb{R}^{n \times n}$, the determinant $\det(A)$ is a scalar value that represents:

• Whether the matrix is invertible:
  • $\det(A) \neq 0$ → invertible
  • $\det(A) = 0$ → not invertible
• The scaling factor of area (2D) or volume (3D) produced by the linear transformation defined by $A$.
• The orientation of space (sign of $\det(A)$):
  • Positive → preserves orientation
  • Negative → reverses orientation

$$\text{If } \det(A) = 0, \text{ the transformation collapses space into a lower dimension.}$$

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🧠 Example in Python (using NumPy)

import numpy as np

# Define a 2x2 matrix
A = np.array([[2, 3],
              [1, 4]])

# Compute the determinant
det_A = np.linalg.det(A)

print(det_A)  # Output: 5.0

$$\det(A) = 2 \cdot 4 - 3 \cdot 1 = 5$$

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✳️ Summary

Concept	Meaning
$\det(A) \neq 0$	Matrix is invertible
$\det(A) = 0$	Matrix is singular
$\det(A)$	Scale factor of area/volume
Sign of $\det(A)$	Indicates orientation