Page 28 - mathematics

Example Properties of matrix addition and scalar
Multiplication:
 9 1 3 4 7 8 ABBA
A  (B  C)  (A  B)  C
If A   2 4 2 , B  9 3 5  , Find 3A-5B.
  (12 )A  1(2A)
7 1 5 1 1 2 
1A  A
Solution 1(A  B)  1A  1B

 27 3 9  20 35 40  (1  2)A  1A  2A
15 25
C  3A   6 12 6  , D  5B   45 10
   5
21 3 15  5

 7 32 31 

3A  5B  C  D  39 3 31
26 8 25

Matrix Multiplication Example

Find the product of the following matrices:

 a11 a12  a1 p   b11 b12  b1n  A  4 7 Β  9 2
 a22  a2 p   b21 3 5 6 8 
 a21   b22  b2n 
AB    am2  
   
  bp1  Solution
am1 amp  bp2  b pn 

 a11b11  ...  a1pbp1  a11b1n  ...  a1pbpn  4 9  7 6 4 (2)  7 8 78 48
   AB 3 9  5 6 3 (2)  5  8 = 57 34
 a21b11  ...  a2 pbp1 a21b1n  ...  a2 pbpn  
  
   

am1b11  ...  ampbp1 am1b1n  ...  ampbpn 

Example Remarks:

Find the product of the following matrices: Am pB pn  Cmn
BA  AB
5 8 B  4 3 A(BC) = (AB)C
A  1 0  2 0  A(B + C) = A B + A C
7 (B + C)A = BA + CA
2

Solution

5  (4)  8  2 5  (3)  0 8 4 15
AB  1 (4)  0  2 4 
1 (3)  0  0   3 
2  (4)  7  2 
2  (3)  0  7  6 6 

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