By Kardi Teknomo, PhD .
LinearAlgebra

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One of the most important matrix operations is matrix multiplication or matrix product. The multiplication of two matrices Matrix Multiplication and Matrix Multiplication is a matrix Matrix Multiplication whose element Matrix Multiplication consists of the vector inner product of the Matrix Multiplication row of matrix Matrix Multiplication and the Matrix Multiplication column of matrix Matrix Multiplication , that is Matrix Multiplication . Matrix multiplication can be done only when the number of columns of Matrix Multiplication is equal to the number of rows of Matrix Multiplication .
If the size of matrix Matrix Multiplication is Matrix Multiplication by Matrix Multiplication and the size of matrix Matrix Multiplication is Matrix Multiplication by Matrix Multiplication , then the result of matrix multiplication is a matrix size Matrix Multiplication by Matrix Multiplication , or in short Matrix Multiplication .

Example
Matrix Multiplication , Matrix Multiplication

Matrix Multiplication
Matrix Multiplication
Matrix Multiplication
Matrix Multiplication


Example
Matrix Multiplication , Matrix Multiplication
Matrix Multiplication
Notice that the matrix multiplication produces null matrix even if the input two matrices are not null matrices.


The interactive program below shows you the result of matrix multiplication. Your input must be two matrices, one with size m by n and the other one with size n by k. Random Example will generate random matrices at the right size. Try to input your own matrices to gain more understanding about matrix product.

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Properties

Some important properties of matrix multiplication are

  • Matrix product is an non-commutative operation. In general, you cannot reverse the order of multiplication Matrix Multiplication . This is very different from multiplication of two numbers.
  • Matrix product is an associative operation. You can exchange the parentheses (to say which order to be computed first) and it does not change the result Matrix Multiplication .
  • Matrix product is a distributive operation. You can distribute (and group) the multiplication with respect to addition as long as the order of multiplication does not change, Matrix Multiplication and Matrix Multiplication
  • Identity matrix is a multiplicative identity matrix such that Matrix Multiplication .
  • Transpose of the product of two matrices is equal to the product of their transposes taken in the reverse order Matrix Multiplication . In general, the transpose of the matrix products can be extended to several matrices Matrix Multiplication
  • Determinant of a matrix multiplication is equal to the multiplication of their determinant, Matrix Multiplication

See also : matrix power , matrix inverse , vector inner product , Matrix element-wise product

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