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Associative Properties of Matrices:

TheAssociative Property of Addition for Matricesstates:

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 ]

TheAssociative Property of Multiplication of Matricesstates:

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

Example 2:

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

Find ( A B ) C and A ( B C ) .

Find ( A B ) C : Find A ( B C ) :

( [ 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 ]