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In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
Others, such as matrix addition, scalar multiplication, matrix multiplication, and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to a field or a ring. [8] In this section, it is supposed that matrix entries belong to a fixed ring, which is typically a field of numbers.
where is the matrix formed by replacing the i-th column of A by the column vector b. A more general version of Cramer's rule [13] considers the matrix equation = where the n × n matrix A has a nonzero determinant, and X, B are n × m matrices.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...
Months after its $80 million Series B fundraise, Course Hero has acquired Symbolab, an artificial intelligence-powered calculator that helps students answer and understand complex math questions.
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]