Determinant
Functions & Advanced AlgebraThe determinant is a scalar value computed from a square matrix that encodes information about the matrix's invertibility and the scaling factor of the transformation it represents.
Formula
\det\begin{pmatrix}a & b\\c & d\end{pmatrix} = ad - bc
Definition
The determinant is a single number calculated from a square matrix. For a $2\times 2$ matrix $\begin{bmatrix}a & b\\c & d\end{bmatrix}$, the determinant is $ad - bc$.
Example
$\det\begin{bmatrix}3 & 1\\2 & 4\end{bmatrix} = (3)(4) - (1)(2) = 12 - 2 = 10$. $\det\begin{bmatrix}2 & 6\\1 & 3\end{bmatrix} = (2)(3) - (6)(1) = 6 - 6 = 0$.
Key Insight
A determinant of $0$ means the matrix has no inverse. Think of the $2\times 2$ determinant as the area of the parallelogram formed by the two row vectors: if the area is $0$, the vectors are parallel and the transformation squishes the plane to a line.
Definition
The determinant of a $2\times 2$ matrix $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ is $\det(A) = ad - bc$. For $3\times 3$ matrices, use cofactor expansion. Key properties: $\det(AB) = \det(A)\det(B)$, $\det(A^T) = \det(A)$, $\det(A^{-1}) = 1/\det(A)$. A matrix is invertible iff $\det(A) \neq 0$.
Example
$\det\begin{bmatrix}1 & 2 & 3\\0 & 4 & 5\\1 & 0 & 6\end{bmatrix}$: expand along row $1$: $1(4\times 6 - 5\times 0) - 2(0\times 6 - 5\times 1) + 3(0\times 0 - 4\times 1) = 24 + 10 - 12 = 22$.
Key Insight
The determinant measures the signed scaling factor of the transformation. $|\det(A)|$ = how volumes are scaled; sign tells orientation. $\det(A) = 0$ means the transformation collapses a dimension (non-invertible).
Definition
The determinant $\det: M_n(F) \to F$ is the unique alternating multilinear form on the columns (or rows) with $\det(I) = 1$. It equals the product of eigenvalues (over an algebraically closed field). Leibniz formula: $\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod a_{i,\sigma(i)}$. The determinant is the top exterior power: $\det(A)$ = the scalar by which $A$ acts on $\Lambda^n(F^n)$.
Example
The Vandermonde determinant: $\det\begin{bmatrix}1 & x_1 & x_1^2 & \ldots\\1 & x_2 & x_2^2 & \ldots\\\vdots & \vdots & \vdots & \ddots\end{bmatrix} = \prod_{i<j} (x_j - x_i)$. This is used in polynomial interpolation, numerical analysis, and algebraic geometry.
Key Insight
The determinant's sign is the parity of the permutation required to sort the rows, connecting linear algebra to the symmetric group $S_n$. The alternating property (swapping two rows negates the determinant) encodes the orientation-preserving/reversing nature of the transformation.