If the symbol
* denotes the addition of twice the first number to three times the second number
(i.e., if a and b are two whole numbers, then a*b = 2a + 3b). Then find the value of:
(i) 5*3
(ii) 106 * 20
(iii) 315 * 25
le 12
1
Answers
Answer:
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Answer:
Let the three numbers be x,y and z
According to the question,
x+y+z=2
x+2y+z=1
5x+y+z=6
The given system of equations can be written in matrix form as follows:
⎣
⎢
⎢
⎡
1
1
5
1
2
1
1
1
1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
2
1
6
⎦
⎥
⎥
⎤
AX=B
Here A=
⎣
⎢
⎢
⎡
1
1
5
1
2
1
1
1
1
⎦
⎥
⎥
⎤
,X=
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
and B=
⎣
⎢
⎢
⎡
2
1
6
⎦
⎥
⎥
⎤
∣A∣=
∣
∣
∣
∣
∣
∣
∣
∣
1
1
5
1
2
1
1
1
1
∣
∣
∣
∣
∣
∣
∣
∣
=1(2−1)−1(1−5)+1(1−10)=1+4−9=−4
Let C
ij
be the co-factors of the elememts a
ij
in A=[a
ij
].Then
C
11
=(−1)
1+1
∣
∣
∣
∣
∣
∣
2
1
1
1
∣
∣
∣
∣
∣
∣
=2−1=1
C
12
=(−1)
1+2
∣
∣
∣
∣
∣
∣
1
5
1
1
∣
∣
∣
∣
∣
∣
=−(1−5)=4
C
13
=(−1)
1+3
∣
∣
∣
∣
∣
∣
1
5
2
1
∣
∣
∣
∣
∣
∣
=1−10=−9
C
21
=(−1)
2+1
∣
∣
∣
∣
∣
∣
1
1
1
1
∣
∣
∣
∣
∣
∣
=1−1=0
C
22
=(−1)
2+2
∣
∣
∣
∣
∣
∣
1
5
1
1
∣
∣
∣
∣
∣
∣
=1−5=−4
C
23
=(−1)
2+3
∣
∣
∣
∣
∣
∣
1
5
1
1
∣
∣
∣
∣
∣
∣
=−(1−5)=4
C
31
=(−1)
3+1
∣
∣
∣
∣
∣
∣
1
2
1
1
∣
∣
∣
∣
∣
∣
=1−2=−1
C
32
=(−1)
3+2
∣
∣
∣
∣
∣
∣
1
1
1
1
∣
∣
∣
∣
∣
∣
=1−1=0
C
33
=(−1)
3+31
∣
∣
∣
∣
∣
∣
1
1
1
2
∣
∣
∣
∣
∣
∣
=2−1=1
adjA=
⎣
⎢
⎢
⎡
1
0
−1
4
−4
0
−9
4
1
⎦
⎥
⎥
⎤
T
=
⎣
⎢
⎢
⎡
1
4
−9
0
−4
4
−1
0
1
⎦
⎥
⎥
⎤
⇒A
−1
=
∣A∣
1
adjA
=
4
−1
⎣
⎢
⎢
⎡
1
4
−9
0
−4
4
−1
0
1
⎦
⎥
⎥
⎤
X=A
−1
B
⇒X=−
4
1
⎣
⎢
⎢
⎡
1
4
−9
0
−4
4
−1
0
1
⎦
⎥
⎥
⎤
⎣
⎢
⎢
⎡
2
1
6
⎦
⎥
⎥
⎤
⇒
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=−
4
1
⎣
⎢
⎢
⎡
2+0−6
8−4+0
−18+4+6
⎦
⎥
⎥
⎤
⇒
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
=−
4
1
⎣
⎢
⎢
⎡
−4
4
−8
⎦
⎥
⎥
⎤
∴x=1,y=−1 and z=2