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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

Matrix Multiplication

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 Multiplicationand Matrix Multiplicationis a matrix Matrix Multiplication whose element Matrix Multiplication consists of the vector inner product of the Matrix Multiplicationrow of matrix Matrix Multiplication and theMatrix Multiplication column of matrixMatrix Multiplication, that isMatrix Multiplication. Matrix multiplication can be done only when the number of columns of Matrix Multiplication is equal to the number of rows ofMatrix Multiplication.
If the size of matrix Matrix Multiplicationis Matrix MultiplicationbyMatrix Multiplication and the size of matrix Matrix Multiplicationis Matrix MultiplicationbyMatrix Multiplication, then the result of matrix multiplication is a matrix size Matrix MultiplicationbyMatrix Multiplication, or in shortMatrix 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 multiplicationMatrix 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 resultMatrix 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 Multiplicationand Matrix Multiplication
  • Identity matrix is a multiplicative identity matrix such thatMatrix Multiplication.
  • Transpose of the product of two matrices is equal to the product of their transposes taken in the reverse orderMatrix 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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Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
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