Matrix Determinant
The determinant is a mathematical property of a square matrix (matrix with equal number of columns and rows) that provides valuable information about the matrix. The determinant is denoted as det(A) or |A|, where A represents the matrix. The determinant is a single value that can be positive, negative, or zero.
The determinant carries several important properties and interpretations:
- Invertibility: A square matrix A is invertible (non-singular) if and only if its determinant is nonzero;
- Area or Volume Scaling: For 2x2 and 3x3 matrices, the determinant provides information about the scaling factor or the change in area/volume under a linear transformation represented by the matrix;
- Linear Independence: The determinant can determine whether a set of vectors is linearly independent. If the determinant of a matrix composed of vectors is nonzero, the vectors are linearly independent;
- Solution Existence: In systems of linear equations represented by matrices, the determinant can determine whether a unique solution exists. If the determinant is nonzero, a unique solution exists; otherwise, there may be no solution or an infinite number of solutions.
In Python, we can calculate determinant using np.linalg.det() method:
import numpy as np
# Define a square matrix
A = np.array([[3, 1], [2, 4]])
# Calculate the determinant
det_A = np.linalg.det(A)
# Print the determinant
print(round(det_A, 2))
import numpy as np # Define a square matrix A = np.array([[3, 1], [2, 4]]) # Calculate the determinant det_A = np.linalg.det(A) # Print the determinant print(round(det_A, 2))
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Section 2. Chapter 3
