Mathematics > Linear Algebra > Determinants

Determinants are a fundamental concept within the broader field of linear algebra, a branch of mathematics that deals with vectors, vector spaces, and linear transformations. The determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix and the linear system it represents.

In more formal terms, if \( A \) is an \( n \times n \) matrix, the determinant of \( A \), denoted as \( \det(A) \) or \( |A| \), is a unique number that is defined recursively using the elements of \( A \). For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the determinant is given by:
\[ \det(A) = ad - bc \]

For larger matrices, the determinant is computed through a process called cofactor expansion (or Laplace expansion), which expresses the determinant of an \( n \times n \) matrix in terms of the determinants of smaller \((n-1) \times (n-1)\) matrices. For instance, the determinant of a 3x3 matrix \( A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \) can be computed as:
\[ \det(A) = a_{11} \det\begin{pmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{pmatrix} - a_{12} \det\begin{pmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{pmatrix} + a_{13} \det\begin{pmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} \]

Determinants have several key properties and applications:
1. Invertibility: A matrix \( A \) is invertible (i.e., it has an inverse) if and only if \( \det(A) \neq 0 \). If \( \det(A) = 0 \), the matrix is singular and does not have an inverse.
2. Volume Interpretation: In geometry, the absolute value of the determinant of a matrix can be interpreted as the scaling factor of the volume when the matrix is viewed as a linear transformation. For example, the determinant of a 2x2 matrix represents the area scaling factor of a parallelogram, and for a 3x3 matrix, it represents the volume scaling factor of a parallelepiped.
3. Eigenvalues: The determinant of a matrix is also related to its eigenvalues. For a matrix \( A \), the determinant is the product of its eigenvalues: if \( A \) has eigenvalues \( \lambda_1, \lambda_2, \ldots, \lambda_n \), then
\[ \det(A) = \lambda_1 \lambda_2 \cdots \lambda_n \]
4. Systems of Linear Equations: Determinants play a crucial role in solving systems of linear equations. They are used in Cramer’s rule, which provides an explicit formula for the solution of a system of linear equations with as many equations as unknowns.

Cramer’s Rule states that for a system of linear equations \( AX = B \), where \( A \) is an \( n \times n \) matrix, \( X \) is the column vector of variables, and \( B \) is the column vector of constants, the solution \( X = (x_1, x_2, \ldots, x_n) \) can be found as:
\[ x_i = \frac{\det(A_i)}{\det(A)} \]
where \( A_i \) is the matrix formed by replacing the \( i \)-th column of \( A \) with the column vector \( B \).

In summary, determinants are an essential tool in linear algebra with wide-ranging applications in mathematics and related fields. Understanding their properties and computational methods is crucial for solving various problems involving matrices and linear transformations.