Determinant
The 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 $\begin{bmatrix}a & b\\c & d\end{bmatrix}$, it is $\det(A) = ad - bc$, and a determinant of $0$ means the matrix has no inverse. Key properties: $\det(AB) = \det(A)\det(B)$, $\det(A^T) = \det(A)$, $\det(A^{-1}) = 1/\det(A)$, and a matrix is invertible if and only if $\det(A) \neq 0$; for $3\times 3$ and larger matrices, cofactor expansion is used. Formally, $\det: M_n(F) \to F$ is the unique alternating multilinear form on the columns (or rows) with $\det(I) = 1$, equal to the product of eigenvalues over an algebraically closed field; the Leibniz formula gives $\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod a_{i,\sigma(i)}$, and geometrically $\det(A)$ is the scalar by which $A$ acts on the top exterior power $\Lambda^n(F^n)$.
Example
$\det\begin{bmatrix}3 & 1\\2 & 4\end{bmatrix} = (3)(4) - (1)(2) = 10$, while $\det\begin{bmatrix}2 & 6\\1 & 3\end{bmatrix} = 6 - 6 = 0$, indicating that matrix has no inverse. Expanding $\det\begin{bmatrix}1 & 2 & 3\\0 & 4 & 5\\1 & 0 & 6\end{bmatrix}$ along row $1$ gives $1(24) - 2(-5) + 3(-4) = 22$. The Vandermonde determinant, $\det\begin{bmatrix}1 & x_1 & x_1^2 & \ldots\\1 & x_2 & x_2^2 & \ldots\\\vdots & \vdots & \vdots & \ddots\end{bmatrix} = \prod_{i<j} (x_j - x_i)$, is used in polynomial interpolation, numerical analysis, and algebraic geometry.
Key Insight
Think of the $2\times 2$ determinant as the area of the parallelogram formed by the two row vectors: if that area is $0$, the vectors are parallel and the transformation squishes the plane to a line, and more generally $|\det(A)|$ measures how volumes are scaled while the sign tells orientation, with $\det(A) = 0$ meaning the transformation collapses a dimension. The determinant's sign is the parity of the permutation required to sort the rows, connecting linear algebra to the symmetric group $S_n$, and the alternating property (swapping two rows negates the determinant) encodes whether the transformation preserves or reverses orientation.