Paul McWhorter pgm7865
Math 311 Problem: 2.2.5

http://208.166.239.244/2_2_5.htm

Verify the associative law of matrix multiplication by calculating the
products of these matrices in two ways.

(a) (7 2) (-1 3 1) (4 7)
(3 1) ( 3 -1 0) (3 5)
(0 0)

(b) (2 3 1) ( 15 20 8) (-6 4 1)
(3 4 1) (-11 -15 -7) ( 5 -3 -1)
(1 2 2) ( 5 8 6) (-2 1 1)

Solution: Parts (a) and (b) may be solved in the same way.
The associative law of matrix multiplication states:

(AB)C = A(BC) = ABC

A set of matrices to be multiplied may be associated with each other in
any order, so long as the original order of the matrices to be multiplied
does not change.

To verify this, we simply assign each matrix a letter name A, B, or C with
respect to the order ib wicth they appear and show that:

(AB)C = A(BC)

(a)

A = (7 2) B = (-1 3 1) C = (4 7)
(3 1) ( 3 -1 0) (3 5)
(0 0)

AB = (-7+6 21-2 7+0) = (-1 19 7)
( 3+3 9-1 3+0) ( 0 8 3)

(AB)C = (-4+57+0 -7+95+0) = (53 88) = A(BC)
( 0+24+0 0+40+0) (24 40)

BC = (-4+9+0 -7+15+0) = (5 8)
(12-3+0 21-5+0) (9 16)

A(BC) = (35+18 56+32) = (53 88) = A(BC)
( 15+9 24+16) (24 40)

(b)

A = (2 3 1) B = ( 15 20 8) C = (-6 4 1)
(3 4 1) (-11 -15 -7) ( 5 -3 -1)
(1 2 2) ( 5 8 6) (-2 1 1)

AB = ( 30-33+5 40-45+8 16-21+6) = (2 3 1)
( 45-44+5 60-60+8 24-28+6) (6 8 2)
(15-22+10 20-30+16 8-14+12) (3 6 6)

(AB)C = ( -12+15-2 8-9+1 2-3+1) = (1 0 0) = A(BC)
( -36+40-4 24-24+2 6-8-2) (0 2 0)
(-18+30-12 12-18+6 3-6+6) (0 0 3)

BC = (-90+100-16 60-60+8 15-20+8) = (-6 8 3)
( 66-75+14 -44+45-7 -11+15-7) ( 5 -6 -3)
( -30+40-12 20-24+6 5-8+6) (-2 2 3)

A(BC) = (-12+15-2 16-18+2 6-9+3) = (1 0 0) = (AB)C
(-18+20-2 24-24+2 9-12+3) (0 2 0)
( -6+10-4 8-12+4 3-6+6) (0 0 3)