Matrix Multiplication


The Dot Product for Vectors in Rn


        a  =  [a1 a2 ... an]

 be a vector in Rn (considered as a 1 by n matrix) and let    

then the dot product of a and b is defined by

        a .  b  =  a1b1 + a2b2 + ... + anbn  = S aibi



Find the dot product of

       a  =  [2  1  0  6  -1]   and    b  =  


We have

        a . b  =  (2)(5) + (1)(2) + (0)(-3) + (6)(0) + (-1)(1)  =  11


Matrix Multiplication

There are many ways of thinking about a matrix.  One way is as a collection of row vectors and another way is as a collection of column vectors.  Consider the m by p matrix A (considered as a matrix of row vectors) and the p by n matrix B (considered as a matrix of column vectors).  The matrices are shown below.


We define the matrix product by

        (AB)ij  =  vi . wj

Remark: If the number of columns of A is not equal to the number of rows of B, then the product AB is not defined.

Remark: It is not true in general that AB and BA are the same matrix even if they are both defined.

We can also define





Then the matrix product is


Linear Systems

Any m by n linear system can be written in the form 

        Ax  =  b

Where A is the coefficient matrix, 

        xT  =  (x1  x2  ...  xn)

and b is the m by 1 matrix of numbers to the left of the equality.  For example the linear system

        2x + 3y + z  =  0
        3x - 4y - z  =  6
        x  + 2y + 3x  =  2

can be written as 


Often, we write the matrix equation in augmented form as shown below


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