Determinants
1. Introduction to Determinants
A determinant is a scalar value associated with a square matrix. It provides important information about the matrix, such as invertibility, and plays a crucial role in linear algebra.
Notation
For a square matrix \(A\):
Determinants of Small Matrices
2x2 Matrix
For \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\):
3x3 Matrix
For \(A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\):
Applications of Determinants
- Determining invertibility (\(\det(A) \neq 0\) implies \(A\) is invertible).
- Calculating volumes of parallelepipeds.
- Solving systems of linear equations using Cramer’s Rule.
- Analyzing the effects of linear transformations.
2. Properties of Determinants
Determinants have several useful properties that simplify calculations and provide insight into matrix behavior.
Key Properties
-
Determinant of Identity Matrix:
\[ \det(I) = 1 \] -
Row or Column Swapping: Swapping two rows (or columns) changes the sign of the determinant.
-
Scalar Multiplication: If a row (or column) is multiplied by a scalar \(k\):
\[ \det(A') = k \det(A) \] -
Additive Property: Adding a multiple of one row to another does not change the determinant.
-
Zero Row or Column: If a matrix has a row or column of all zeros:
\[ \det(A) = 0 \] -
Upper or Lower Triangular Matrix: For triangular matrices, the determinant is the product of the diagonal elements:
\[ \det(A) = a_{11}a_{22}\cdots a_{nn} \]
Code Example
import numpy as np
# Define a matrix
A = np.array([[2, 1], [5, 3]])
# Compute the determinant
det_A = np.linalg.det(A)
print("Determinant:", round(det_A))
3. Cramer’s Rule, Volume, and Linear Transformations
Cramer’s Rule
Cramer’s Rule solves systems of linear equations \(A\mathbf{x} = \mathbf{b}\) using determinants.
Formula
For \(n \times n\) matrix \(A\):\
Where \(A_i\) is the matrix formed by replacing the \(i\)-th column of \(A\) with \(\mathbf{b}\).
Example
Solve:
Solution:
Code Example
# Define matrices
A = np.array([[2, 3], [4, 1]])
b = np.array([8, 10])
# Solve using Cramer's Rule
det_A = np.linalg.det(A)
x1 = np.linalg.det(np.column_stack((b, A[:, 1]))) / det_A
x2 = np.linalg.det(np.column_stack((A[:, 0], b))) / det_A
print("Solution:", (x1, x2))
Volume and Determinants
The determinant of a matrix formed by vectors \(\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\) represents the volume of the parallelepiped they span.
Formula
For vectors \(\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\):
Example
Calculate the volume of a parallelepiped spanned by:
Solution:
Code Example
# Define vectors
v1 = np.array([1, 0, 0])
v2 = np.array([0, 1, 0])
v3 = np.array([0, 0, 2])
# Form matrix and compute volume
A = np.column_stack((v1, v2, v3))
volume = abs(np.linalg.det(A))
print("Volume:", volume)
Determinants in Linear Transformations
The determinant of a transformation matrix \(T\) indicates how the transformation scales areas or volumes:
- \(\det(T) > 0\): Preserves orientation.
- \(\det(T) < 0\): Reverses orientation.
- \(\det(T) = 0\): Flattens the space (non-invertible).
This guide covers determinants in-depth with examples, properties, and practical applications. Let me know if you need additional sections or clarifications!