A
3
2
1
3
2
1
1
12
8
4
2
1
4
2
1
14
6
15

 




 



=
=
−
−
=

 




 




 



Rationalised 2023-24
MATRICES         57
So
A3 = A A2 = 
1
2
3
19
4
8
63
46
69
3
2
1
1
12
8
69
6
23
4
2
1
14
6
15
92
46
63

 




 



−
=
−

 




 




 



Now
A3 – 23A – 40I = 
63
46
69
1
2
3
1
0
0
69
6
23 – 23 3
2
1 – 40 0
1
0
92
46
63
4
2
1
0
0 1












−
−


















=
63
46
69
23
46
69
40
0
0
69
6
23
69
46
23
0
40
0
92
46
63
92
46
23
0
0
40
−
−
−
−












−
+ −
−
+
−












−
−
−
−






=
63
23
40
46
46
0
69
69
0
69
69
0
6
46
40
23
23
0
92
92
0
46
46
0
63
23
40
−
−
−
+
−
+




−
+
− +
−
−
+




−
+
−
+
−
−


=
0
0
0
0
0
0
O
0
0
0



 =






Example 19 In a legislative assembly election, a political group hired a public relations
firm to promote its candidate in three ways: telephone, house calls, and letters The
cost per contact (in paise) is given in matrix A as
A = 
40
Telephone
100
Housecall
50
Letter
Cost per contact









The number of contacts of each type made in two cities X and Y is given by
Telephone
Housecall
Letter
1000
500
5000
X
B
Y
3000
1000 10,000
→


= 
→

 Find the total amount spent by the group in the two
cities X and Y Rationalised 2023-24
 58
MATHEMATICS
Solution We have
BA =
40,000
50,000
250,000
X
Y
120,000 +100,000 +500,000
+
+
→



 →


=
340,000
X
Y
720,000
→



 →


So the total amount spent by the group in the two cities is 340,000 paise and
720,000 paise, i