Multiplication of Matrices

If A = [a_ij] is an m x n matrix and B = [b_ij] is an n x p matrix, then the product AB is an m x p matrix.

Sample Problem #1

       [   1   5   2   ]          [ 3 ]

A=   [   2   9   5   ]    B= [ 5 ]

       [   5   2   6   ]          [ 1 ]

Matrix A is measured to be 3 x 3, while Matrix B is 3 x 1.

This means that we can multiply the the matrices with each other.

This Results to

[ (1)(3) + (5)(5) + (2)(1) ]

[ (2)(3) + (9)(5) + (5)(1) ]

[ (5)(3) + (2)(5) + (6)(1) ]

   [ 3 + 25 + 2 ]

= [ 6 + 45 + 5 ]

   [ 15 + 10 +6]

    [ 30 ]

=  [ 56 ]

    [ 31 ]

Sample Problem #2

       [ 2  7  5 ]

A = [ 1  3  6 ]       B = [ 3  1  8 ]

       [ 3  2  1 ]

Since Matrix A is measured to be 3 x 3, and Matrix B is  1 x 3, we cannot multiply the matrices with each other. Therefore, it is undefined.