Associative Properties of Matrices:

The Associative Property of Addition for Matrices states:

Let A , B and C be m×n matrices.  Then, ( A+B )+C=A+( B+C ) .

Example 1:

A=[ 3 2 4 1 0 5 ],B=[ 2 3 1 4 2 0 ],C=[ 8 1 5 6 1 2 ]

Find ( A+B )+C and A+( B+C )

        Find ( A+B )+C :      

         ( [ 3 2 4 1 0 5 ]+[ 2 3 1 4 2 0 ] )+[ 8 1 5 6 1 2 ] =[ 1 5 3 3 2 5 ]+[ 8 1 5 6 1 2 ] =[ 9 4 8 9 3 3 ]

        Find A+( B+C ) :

         [ 3 2 4 1 0 5 ]+( [ 2 3 1 4 2 0 ]+[ 8 1 5 6 1 2 ] ) =[ 3 2 4 1 0 5 ]+[ 6 2 4 10 3 2 ] =[ 9 4 8 9 3 3 ]

The Associative Property of Multiplication of Matrices states:

Let A , B and C be n×n matrices.  Then, ( AB )C=A( BC )

Example 2:

A=[ 3 2 1 0 ],B=[ 2 3 4 2 ],C=[ 1 5 1 2 ]

Find ( AB )C and A( BC ) .

         Find ( AB )C :                                                      Find A( BC ) :

          ( [ 3 2 1 0 ][ 2 3 4 2 ] )[ 1 5 1 2 ] =[ 2 13 2 3 ][ 1 5 1 2 ] =[ 11 36 5 4 ] [ 3 2 1 0 ]( [ 2 3 4 2 ][ 1 5 1 2 ] ) =[ 3 2 1 0 ][ 5 4 2 24 ] =[ 11 36 5 4 ]