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  2. Determinant - Wikipedia

    en.wikipedia.org/wiki/Determinant

    The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the Chinese mathematics textbook The Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the

  3. Rule of Sarrus - Wikipedia

    en.wikipedia.org/wiki/Rule_of_Sarrus

    Rule of Sarrus: The determinant of the three columns on the left is the sum of the products along the down-right diagonals minus the sum of the products along the up-right diagonals.

  4. Leibniz formula for determinants - Wikipedia

    en.wikipedia.org/wiki/Leibniz_formula_for...

    Instead, the determinant can be evaluated in () operations by forming the LU decomposition = (typically via Gaussian elimination or similar methods), in which case = and the determinants of the triangular matrices and are simply the products of their diagonal entries. (In practical applications of numerical linear algebra, however, explicit ...

  5. Jacobian matrix and determinant - Wikipedia

    en.wikipedia.org/wiki/Jacobian_matrix_and...

    In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.

  6. Cramer's rule - Wikipedia

    en.wikipedia.org/wiki/Cramer's_rule

    In the 2×2 case, if the coefficient determinant is zero, then the system is incompatible if the numerator determinants are nonzero, or indeterminate if the numerator determinants are zero. For 3×3 or higher systems, the only thing one can say when the coefficient determinant equals zero is that if any of the numerator determinants are nonzero ...

  7. Minor (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Minor_(linear_algebra)

    Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n.A k × k minor of A, also called minor determinant of order k of A or, if m = n, the (n − k) th minor determinant of A (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns.

  8. Jacobi's formula - Wikipedia

    en.wikipedia.org/wiki/Jacobi's_formula

    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

  9. Hadamard's inequality - Wikipedia

    en.wikipedia.org/wiki/Hadamard's_inequality

    In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants [1]) is a result first published by Jacques Hadamard in 1893. [2] It is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors.