Advertisements
Advertisements
Question
The cost of 2 Kg of Wheat and 1 Kg of Sugar is ₹ 70. The cost of 1 Kg of Wheat and 1 Kg of Rice is ₹ 70. The cost of 3 Kg of Wheat, 2 Kg of Sugar and 1 Kg of rice is ₹ 170. Find the cost of per kg each item using the matrix inversion method.
Solution
Let the cost of 1 kg of wheat, sugar and rice be x, y, z respectively.
By given data,
2x + y = 70
x + z = 70
3x + 2y + z = 170
`=> [(2,1,0),(1,0,2),(3,2,1)] [(x),(y),(z)] = [(70),(70),(170)]`
⇒ AX = B
Where A = `[(2,1,0),(1,0,2),(3,2,1)]`, X = `[(x),(y),(z)]`, B = `[(70),(70),(170)]`
⇒ X = A-1B
`|"A"| = |(2,1,0),(1,0,1),(3,2,1)|`
`= 2|(0,1),(2,1)| - 1|(1,1),(3,1)| + 0|(1,0),(3,2)|`
= 2(0 - 2) - 1(1 - 3) + 0
= - 4 + 2 + 0
= - 2 ≠ 0
∴ `"A"^-1` exists
adj A = `[(+|(0,1),(2,1)| -|(1,1),(3,1)| +|(1,0),(3,2)|),
(-|(1,0),(2,1)| + |(2,0),(3,1)| - |(2,1),(3,2)|),(+|(1,0),(0,1)|-|(2,0),(1,1)| +|(2,1),(1,0)|)]^"T"`
`= [(+(0-2),-(1-3),+(2-0)),(-(1-0),+(2-0),-(4-3)),(+(1-0),-(2-0),+(0-1))]^"T"`
`= [(-2,2,2),(-1,2,-1),(1,-2,-1)]^"T"`
`= [(-2,-1,1),(2,2,-2),(2,-1,-1)]`
`"A"^-1 = 1/|"A"|`adj A
`= (-1)/2[(-2,-1,1),(2,2,-2),(2,-1,-1)]`
X = `"A"^-1"B"`
`[(x),(y),(z)] = (-1)/2[(-2,-1,1),(2,2,-2),(2,-1,-1)] [(70),(70),(170)]`
`[(x),(y),(z)] = (-1)/2 [(-140-70+170),(140+140-340),(140-70-170)]`
`[(x),(y),(z)] = (-1)/2 [(-40),(-60),(-100)]`
`[(x),(y),(z)] = [(20),(30),(50)]`
∴ Cost of 1 kg of wheat is Rs. 20.
Cost of 1 kg of Sugar is Rs. 30.
Cost of 1 kg of Rice is Rs. 50.
APPEARS IN
RELATED QUESTIONS
Find the inverse of the following matrix (if they exist):
`[(2,-3),(5,7)]`
Choose the correct answer from the given alternatives in the following question:
If A = `[("cos"alpha, - "sin"alpha,0),("sin"alpha,"cos"alpha,0),(0,0,1)]` where α ∈ R, then [F(α)]-1 is
A = `[(cos alpha, - sin alpha, 0),(sin alpha, cos alpha, 0),(0, 0, 1)]`, then A−1 is
If A = `[(-4, -3, -3),(1, 0, 1),(4, 4, 3)]`, find adj (A).
Complete the following activity to find inverse of matrix using elementary column transformations and hence verify.
`[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` B−1 = `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
C1 → C1 + C3
`[("( )", 0, -1),("( )", 1, 0),("( )", 1, 3)]` B−1 = `[("( )", 0, 0),("( )", 1, 0),("( )", 0, 1)]`
C3 → C3 + C1
`[(1, 0, 0),("( )", 1, "( )"),(3, 1, "( )")]` B−1 = `[(1, 0, "( )"),(0, 1, 0),("( )", 0, "( )")]`
C1 → C1 – 5C2, C3 → C3 – 5C2
`[(1, "( )", 0),(0, 1, 0),("( )", 1, "( )")]` B−1 = `[(1, 0, "( )"),("( )", 1, -5),(1, "( )", 2)]`
C1 → C1 – 2C3, C2 → C2 – C3
`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]` B−1 = `[(3, -1, "( )"),("( )", 6, -5),(5, "( )", "( )")]`
B−1 = `[("( )", "( )", "( )"),("( )", "( )", "( )"),("( )", "( )", "( )")]`
`[(2, "( )", -1),("( )", 1, 0),(0, 1, "( )")] [(3, "( )", "( )"),("( )", 6, "( )"),("( )", -2, "( )")] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
If A is 3 × 3 matrix and |A| = 4 then |A-1| is equal to:
Solve by using matrix inversion method:
x - y + z = 2, 2x - y = 0, 2y - z = 1
If A = `[(1,-1,1),(2,1,-3),(1,1,1)]`, then the sum of the elements of A-1 is ______.
The inverse of the matrix A = `[(3, 0, 0),(0, 4, 0),(0, 0, 5)]` is ______.
For an invertible matrix A, if A (adj A) = `|(20, 0),(0, 20)|`, then | A | = ______.
