Determinant

The determinant is a scalar computed from a square matrix that encodes whether the matrix is invertible and how its linear transformation scales signed volume.

Leibniz formula: A determinant formula that sums signed products formed by choosing one entry from each row and column. It defines the determinant directly through permutations of matrix entries.
Square matrix: A matrix with the same number of rows and columns. Only square matrices have determinants in the standard definition.
Cramer's rule: A formula expressing solutions of a nonsingular linear system as ratios of determinants. It uses determinants to isolate individual coordinates of a system's solution.
Matrix trace: The sum of the diagonal entries of a square matrix. Trace summarizes eigenvalue sums, whereas determinant summarizes their product.
Laplace expansion: A method for computing a determinant by expanding along one row or column using cofactors. It reduces a determinant to smaller determinants, making recursive computation possible.
Linear transformation: A function between vector spaces that preserves addition and scalar multiplication. A matrix represents a linear transformation whose volume scaling the determinant records.
Jacobian matrix: A matrix of first-order partial derivatives describing a multivariable function's local linear approximation. Its determinant gives the function's local signed volume scaling.
Rank of a matrix: The dimension of the column space of a matrix. Rank measures the dimension of the image; determinant only tests full rank for square matrices.
Cofactor: A signed minor associated with a particular entry of a matrix. Cofactors supply the signed terms in determinant expansion.
Invertible matrix: A square matrix that has a two-sided multiplicative inverse. A matrix is invertible exactly when its determinant is nonzero.

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