Determinants | Generated by AI
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Chapter: Determinants
1. Definition of a Determinant
A determinant is a scalar value that can be computed from the elements of a square matrix. It provides essential information about the matrix, such as invertibility, volume scaling, and solutions to linear equations.
For an \( n \times n \) matrix \( A \), the determinant is denoted as:
[ \det(A) \quad \text{or} \quad |A| ]
2. Computation of Determinants
2.1 Determinant of a 2×2 Matrix
For a \( 2 \times 2 \) matrix:
[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} ]
The determinant is given by:
[ \det(A) = ad - bc ]
2.2 Determinant of a 3×3 Matrix
For a \( 3 \times 3 \) matrix:
[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} ]
The determinant can be computed using cofactor expansion along the first row:
[ \det(A) = a_{11} \begin{vmatrix} a_{22} & a_{23} \ a_{32} & a_{33} \end{vmatrix}
- a_{12} \begin{vmatrix} a_{21} & a_{23} \ a_{31} & a_{33} \end{vmatrix}
- a_{13} \begin{vmatrix} a_{21} & a_{22} \ a_{31} & a_{32} \end{vmatrix} ]
Each \( 2 \times 2 \) determinant is computed as shown earlier.
2.3 Determinants of Higher-Order Matrices
For an \( n \times n \) matrix, determinants can be computed using:
- Cofactor Expansion (Laplace expansion along any row or column)
- Row Reduction Method (Transforming into upper triangular form and multiplying the diagonal elements)
- Leibniz Formula (Sum over all permutations, practical only for small \( n \))
3. Properties of Determinants
[ \det(I_n) = 1 ]
[ \det(B) = -\det(A) \quad \text{if } B \text{ is obtained by swapping two rows of } A. ]
[ \det(A) = 0 \quad \text{if a row (or column) is entirely zero.} ]
[ \det(kA) = k^n \det(A) ]
[ \det(A) = \text{Product of diagonal elements} \quad \text{if } A \text{ is triangular.} ]
[ \det(AB) = \det(A) \det(B) ]
[ \det(A^{-1}) = \frac{1}{\det(A)} \quad \text{if } A \text{ is invertible.} ]
[ \det(A^T) = \det(A) ]
4. Applications of Determinants
4.1 Checking Invertibility
A square matrix \( A \) is invertible if and only if
[ \det(A) \neq 0 ]
4.2 Cramer’s Rule (Solving Linear Systems)
For a system \( Ax = b \), where \( A \) is an \( n \times n \) invertible matrix, the solution for each variable \( x_i \) is:
[ x_i = \frac{\det(A_i)}{\det(A)} ]
where \( A_i \) is obtained by replacing the \( i \)-th column of \( A \) with \( b \).
4.3 Volume Interpretation
The absolute value of the determinant of an \( n \times n \) matrix represents the volume of the transformed unit hypercube:
[ \text{Volume} = |\det(A)| ]
4.4 Eigenvalues and Characteristic Equation
The determinant is used in finding eigenvalues via the characteristic equation:
[ \det(A - \lambda I) = 0 ]
This tutorial provides a fundamental understanding of determinants, including their computation, properties, and applications. Let me know if you need further explanations or examples!