The cost of 4kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 70. Find cost of each item per kg by matrix method
The cost of 4kg onion, 3 kg wheat and 2 kg rice is Rs. 60. The cost of 2kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 70. Find cost of each item per kg by matrix method
Let the prices (per kg) of onion, wheat and rice be Rs. x, Rs. y and Rs. z respectively then
4x+3y+2z=60,2x+4y+6z=90,6x+2y+3z=70
This system equations can be written as AX=B, where
A=⎡⎢⎣432246623⎤⎥⎦,X=⎡⎢⎣xyz⎤⎥⎦ and B=⎡⎢⎣609070⎤⎥⎦
Here, |A|=⎡⎢⎣432246623⎤⎥⎦=4(12−12)−36(6−36)+2(4−24)
Thus, A is non-singular, Therefore, its inverse exists. Therefore, the given system is consistent and has a unique solution given by X=A−1B
Cofactors of A are,
A11=12−12=0,A12=−(6−36)=30,A13=(4−24)=−20
A21=−(9−4)=−5,A22=(12−12)=0,A23=−(8−18)=10
A31=(18−8)=10,A32=−(24−4)=−20,A33=(16−6)=10
adj(A)=⎡⎢⎣030−20−501010−2010⎤⎥⎦=⎡⎢⎣0−510300−20−201010⎤⎥⎦
∴ A−1=1|A|(adj A)=150⎡⎢⎣0−510300−20−201010⎤⎥⎦
Now, X=A−1B⇒⎡⎢⎣xyz⎤⎥⎦=150⎡⎢⎣0−510300−20−201010⎤⎥⎦⎡⎢⎣609070⎤⎥⎦
=150⎡⎢⎣0−450+7001800+0−1400−1200+900+700⎤⎥⎦
⇒⎡⎢⎣xyz⎤⎥⎦=150⎡⎢⎣250400400⎤⎥⎦=⎡⎢⎣588⎤⎥⎦⇒⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣588⎤⎥⎦
∴ x=5,y=8 and z=8. Hence, price of onion per kg is Rs. 5, price of wheat per kg is Rs. 8 and that of rice per kg is Rs. 8
Let the prices (per kg) of onion, wheat and rice be Rs. x, Rs. y and Rs. z respectively then
4x+3y+2z=60,2x+4y+6z=90,6x+2y+3z=70
This system equations can be written as AX=B, where
A=⎡⎢⎣432246623⎤⎥⎦,X=⎡⎢⎣xyz⎤⎥⎦ and B=⎡⎢⎣609070⎤⎥⎦
Here, |A|=⎡⎢⎣432246623⎤⎥⎦=4(12−12)−36(6−36)+2(4−24)
Thus, A is non-singular, Therefore, its inverse exists. Therefore, the given system is consistent and has a unique solution given by X=A−1B
Cofactors of A are,
A11=12−12=0,A12=−(6−36)=30,A13=(4−24)=−20
A21=−(9−4)=−5,A22=(12−12)=0,A23=−(8−18)=10
A31=(18−8)=10,A32=−(24−4)=−20,A33=(16−6)=10
adj(A)=⎡⎢⎣030−20−501010−2010⎤⎥⎦=⎡⎢⎣0−510300−20−201010⎤⎥⎦
∴ A−1=1|A|(adj A)=150⎡⎢⎣0−510300−20−201010⎤⎥⎦
Now, X=A−1B⇒⎡⎢⎣xyz⎤⎥⎦=150⎡⎢⎣0−510300−20−201010⎤⎥⎦⎡⎢⎣609070⎤⎥⎦
=150⎡⎢⎣0−450+7001800+0−1400−1200+900+700⎤⎥⎦
⇒⎡⎢⎣xyz⎤⎥⎦=150⎡⎢⎣250400400⎤⎥⎦=⎡⎢⎣588⎤⎥⎦⇒⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣588⎤⎥⎦
∴ x=5,y=8 and z=8. Hence, price of onion per kg is Rs. 5, price of wheat per kg is Rs. 8 and that of rice per kg is Rs. 8
