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  2. Invertible matrix - Wikipedia

    en.wikipedia.org/wiki/Invertible_matrix

    In linear algebra, an invertible matrix is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their ...

  3. Matrix multiplication - Wikipedia

    en.wikipedia.org/wiki/Matrix_multiplication

    Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]

  4. Cancellation property - Wikipedia

    en.wikipedia.org/wiki/Cancellation_property

    Matrix multiplication also does not necessarily obey the cancellation law. If AB = AC and A ≠ 0, then one must show that matrix A is invertible (i.e. has det(A) ≠ 0) before one can conclude that B = C. If det(A) = 0, then B might not equal C, because the matrix equation AX = B will not have a unique solution for a non-invertible matrix A.

  5. Transpose - Wikipedia

    en.wikipedia.org/wiki/Transpose

    If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.

  6. General linear group - Wikipedia

    en.wikipedia.org/wiki/General_linear_group

    In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.

  7. Kronecker product - Wikipedia

    en.wikipedia.org/wiki/Kronecker_product

    In the language of Category theory, the mixed-product property of the Kronecker product (and more general tensor product) shows that the category Mat F of matrices over a field F, is in fact a monoidal category, with objects natural numbers n, morphisms n → m are n×m matrices with entries in F, composition is given by matrix multiplication ...

  8. Hadamard product (matrices) - Wikipedia

    en.wikipedia.org/wiki/Hadamard_product_(matrices)

    The identity matrix under Hadamard multiplication of two m × n matrices is an m × n matrix where all elements are equal to 1. This is different from the identity matrix under regular matrix multiplication, where only the elements of the main diagonal are equal to 1. Furthermore, a matrix has an inverse under Hadamard multiplication if and ...

  9. Matrix (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Matrix_(mathematics)

    A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group. [65] [66] Since a group of every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups.