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There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending ...
The determinant of the left hand side is the product of the determinants of the three matrices. Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1. The determinant of the middle matrix is our desired value. The determinant of the right hand side is simply (1 + v T u). So we have the result:
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [ 1 ] If A is a differentiable map from the real numbers to n × n matrices, then
When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]
In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a matrix named after the French mathematician Pierre Frédéric Sarrus. [ 1 ] Consider a 3 × 3 {\displaystyle 3\times 3} matrix
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A {\displaystyle A} is an n × n {\displaystyle n\times n} matrix, where a i j {\displaystyle a_{ij}} is the entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th ...
If n = m, the case where A and B are square matrices, ([]) = {[]} (a singleton set), so the sum only involves S = [n], and the formula states that det(AB) = det(A)det(B). For m = 0, A and B are empty matrices (but of different shapes if n > 0), as is their product AB ; the summation involves a single term S = Ø, and the formula states 1 = 1 ...
More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if R T = R −1 and det R = 1. The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO( n ) , one example of which is ...