The Determinant

The Determinant#

The determinant is a fundamental concept in linear algebra that associates a single scalar value with every square matrix. This value, denoted as \(\det(A)\) or \(|A|\), encapsulates important properties of the matrix and the linear transformation it represents. It can be uniquely defined by four key properties.

  1. Identity Matrix: The determinant of the identity matrix is 1, i.e., \(\det(I) = 1\).

  2. Column Swaps: Exchanging two columns of a matrix multiplies its determinant by -1.

  3. Scalar Multiplication: Multiplying a single column by a scalar \(\alpha\) multiplies the entire determinant by \(\alpha\).

  4. Column Operations: Adding a multiple of one column to another column does not change the determinant.

The Leibniz Formula#

While the properties above define the determinant, it can be calculated explicitly using the Leibniz formula. This formula sums over all possible permutations of the matrix’s column indices.

\[ \det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) a_{1,\sigma(1)} \cdots a_{n,\sigma(n)} \]

Here, \(S_n\) is the set of all permutations of the numbers \(\{1, 2, \dots, n\}\), and \(\text{sgn}(\sigma)\) is the signature of the permutation \(\sigma\). The signature is +1 if the permutation can be formed by an even number of swaps and -1 if it requires an odd number of swaps.

Key Properties#

The determinant has several crucial properties that are used throughout mathematics and its applications.

  • Transpose: The determinant of a matrix is equal to the determinant of its transpose: \(\det(A) = \det(A^T)\).

  • Multiplicativity: The determinant of a product of matrices is the product of their determinants: \(\det(AB) = \det(A)\det(B)\).

  • Inverse: The determinant of an inverse matrix is the reciprocal of the original determinant: \(\det(A^{-1}) = \det(A)^{-1}\).

  • Singularity: A square matrix \(A\) is singular (i.e., not invertible) if and only if its determinant is zero: \(\det(A) = 0\).

  • Triangular Matrices: If \(A\) is a triangular (upper or lower) matrix, its determinant is simply the product of its diagonal entries: \(\det(A) = \prod_{i=1}^n a_{ii}\).

Geometric Interpretation 📐#

Geometrically, the determinant represents the signed volume of the \(n\)-dimensional parallelepiped formed by the column vectors of the matrix.

  • The absolute value of the determinant, \(|\det(A)|\), gives the scaling factor of the transformation. For instance, in 2D, it’s the area of the parallelogram formed by the column vectors. In 3D, it’s the volume of the parallelepiped.

  • The sign of the determinant indicates whether the transformation preserves or reverses the orientation of space. A positive determinant preserves orientation (like a rotation), while a negative determinant reverses it (like a reflection).

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