Page 17 - Top Class F5 - Mathematics (Chapter 2)

Mathematics Form 5 Chapter 2 Matrices
19. Find the products of matrices, FG and GF. Hence, determine whether matrix G is an identity matrix. PL 2
Cari hasil darab matriks, FG dan GF. Seterusnya, tentukan sama ada matriks G ialah matriks identiti.
Example –4 3 1 0
(a) F =   , G =  
 5 –1    1 2 0 1
1 0
F = 7 9 , G = 1 1
  
 –4 3 1 0 = –4 3 
1 2
 5 –1 1 0 = 16 9  If AB = BA = A, then B is     –4 3 
1 2 0 1
  
4 –1
7 9 1 1
1 0 –4 3
5 –1
1 0 5 –1
0 1 1 2
    12 8  an identity matrix. FG = GF = F = 1 2
Jika AB = BA = A, maka B
=
1 1 7 9
ialah matriks identiti.
FG  GF  F Hence, G is an identity matrix.
Hence, G is not an identity matrix. Maka, G ialah matriks identiti.
Maka, G bukan matriks identiti.
 2 4 5     8 –3 –1   
0 0 1
1 0 0
(b) F = –5 1 7 , G = 0 1 0 (c) F = 2 0 –6 , G = 0 1 0
3 6 –1 1 0 0 1 4 5 0 0 1
2 4 5 0 0 1 5 4 2 8 –3 –1 1 0 0 8 –3 –1
         
–5 1 7 0 1 0 = 7 1 –5 2 0 –6 0 1 0 = 2 0 –6
3 6 –1 1 0 0 –1 6 3 1 4 5 0 0 1 1 4 5
    3 6 –1      8 –3 –1 
1 0 0 8 –3 –1
0 0 1 2 4 5
0 1 0 –5 1 7 = –5 1 7 0 1 0 2 0 –6 = 2 0 –6
1 0 0 3 6 –1 2 4 5 0 0 1 1 4 5 1 4 5
FG  GF  F FG = GF = F
Hence, G is not an identity matrix. Hence, G is an identity matrix.
Maka, G bukan matriks identiti. Maka, G ialah matriks identiti.

2 9
 5 1   
20. Given that K = 2 –3 and L = 6 4 , solve each of the following. PL 3
2 9
 5 1   
Diberi K = 2 –3 dan L = 6 4 , selesaikan setiap yang berikut.

Example (a) (K + L)I
KI – IL 5 1 2 9 1 0
 2 –3    
+
6 4 0 1
    
 5 1 1 0 – 1 0 2 9  7 10 1 0
 
0 1 6 4
2 –3 0 1
 5 1    = 8 1 0 1
2 9
= 2 –3 – 6 4 7 10
 3 –8  =  8 1 
= –4 –7
(b) 2IL + I (c) (IK)L
2
2 9
1 0 5 1
1 0 1 0
1 0 2 9
2          
+
0 1 2 –3 6 4
0 1 0 1
0 1 6 4
 
    =  5 1 2 9
2 9
1 0
= 2 6 4 + 0 1 2 –3 6 4
 5 18  =  16 49 
= 12 9 –14 6
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