Advertisements
Advertisements
Question
Solve the following equations by method of inversion.
x + y + z = 1, x – y + z = 2 and x + y – z = 3
Solution
Matrix form of the given system of equations is
`[(1,1,1),(1, -1, 1),(1, 1, -1)][(x),(y),(z)] = [(1),(2),(3)]`
This is of the form AX = B,
where A = `[(1,1,1),(1, -1, 1),(1, 1, -1)], "X" = [(x),(y),(z)] "and B"= [(1),(2),(3)]`
Consider AA–1 = I
∴ `[(1,1,1),(1, -1, 1),(1, 1, -1)],"A"^-1 = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
Appying R2 → R2 – R1 and R3 → R3 – R1, we get
`[(1, 1, 1),(0, -2, 0),(0, 0, -2)]"A"^-1 = [(1, 0, 0),(-1, 1, 0),(-1, 0, 1)]`
Appying R2 → `(-1/2)` R2, we get
`[(1, 1, 1),(0, 1, 0),(0, 0, -2)]"A"^-1 = [(1, 0, 0),(1/2, -1/2, 0),(-1, 0, 1)]`
Appying R1 → R1 – R2, we get
`[(1, 0, 1),(0, 1, 0),(0, 0, -2)]"A"^-1 = [(1/2, 1/2, 0),(1/2, -1/2, 0),(-1, 0, 1)]`
Appying R3 → `(-1/2)` R3, we get
`[(1, 0, 1),(0, 1, 0),(0, 0, 1)]"A"^-1 = [(1/2, 1/2, 0),(-1/2, -1/2, 0),(-1/2, 0, -1/2)]`
Appying R1 → R1 – R3, we get
`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]"A"^-1 = [(0, 1/2, 1/2),(1/2, -1/2, 0),(1/2, 0, -1/2)]`
∴ A–1 = `[(0, 1/2, 1/2),(1/2, -1/2, 0),(1/2, 0, -1/2)]`
pre-multiplying AX = B by A–1, we get
A–1(AX) = A–1B
∴ (A–1A)X = A–1B
∴ IX = A–1B
∴ X = A–1B
∴ `[(x),(y),(z)] = [(0, 1/2, 1/2),(1/2, -1/2, 0),(1/2, 0, -1/2)][(1),(2),(3)]`
∴ `[(x),(y),(z)] = [(0 + 1 + 3/2),(1/2 - 1 + 0),(1/2 + 0 - 3/2)] = [(5/2),(-1/2),(-1)]`
∴ By equality of matrices, we get
x = `(5)/(2), y = -(1)/(2) "and z = -1`.
APPEARS IN
RELATED QUESTIONS
Solve the following equations by the reduction method.
3x – y = 1, 4x + y = 6
Solve the following equations by the reduction method.
5x + 2y = 4, 7x + 3y = 5
Solve the following equation by the method of inversion:
2x - y = - 2, 3x + 4y = 3
Solve the following equation by the method of inversion:
5x − y + 4z = 5, 2x + 3y + 5z = 2 and 5x − 2y + 6z = −1
The cost of 4 pencils, 3 pens, and 2 books is ₹ 150. The cost of 1 pencil, 2 pens, and 3 books is ₹ 125. The cost of 6 pencils, 2 pens, and 3 books is ₹ 175. Find the cost of each item by using matrices.
An amount of ₹ 5000 is invested in three types of investments, at interest rates 6%, 7%, 8% per annum respectively. The total annual income from these investments is ₹ 350. If the total annual income from the first two investments is ₹ 70 more than the income from the third, find the amount of each investment using matrix method.
Solve the following equations by method of inversion.
x + 2y = 2, 2x + 3y = 3
Solve the following equations by method of inversion.
2x + y = 5, 3x + 5y = – 3
Express the following equations in matrix form and solve them by method of reduction.
x + 3y = 2, 3x + 5y = 4
The total cost of 3 T.V. and 2 V.C.R. is ₹ 35,000. The shopkeeper wants profit of ₹1000 per television and ₹ 500 per V.C.R. He can sell 2 T.V. and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. and a V.C.R.
The sum of the cost of one Economic book, one Co-operation book and one account book is ₹ 420. The total cost of an Economic book, 2 Co-operation books and an Account book is ₹ 480. Also the total cost of an Economic book, 3 Co-operation books and 2 Account books is ₹ 600. Find the cost of each book using matrix method.
If x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6, then (y, z) = _______
Solve the following :
Two farmers Shantaram and Kantaram cultivate three crops rice, wheat and groundnut. The sale (in Rupees) of these crops by both the farmers for the month of April and May 2016 is given below,
| April 2016 (in ₹.) | |||
| Rice | Wheat | Groundnut | |
| Shantaram | 15000 | 13000 | 12000 |
| Kantaram | 18000 | 15000 | 8000 |
| May 2016 (in ₹.) | |||
| Rice | Wheat | Groundnut | |
| Shantaram | 18000 | 15000 | 12000 |
| Kantaram | 21000 | 16500 | 16000 |
Find : The total sale in rupees for two months of each farmer for each crop.
Solve the following equations by method of inversion : x + y – z = 2, x – 2y + z = 3 and 2x – y – 3z = – 1
If A2 + 5A + 3I = 0, |A| ≠ 0, then A–1 = ______
State whether the following statement is True or False:
If O(A) = m × n and O(B) = n × p with m ≠ p, then BA exists but AB does not exist.
If the volume of the parallelepiped whose conterminus edges are along the vectors a, b, c is 12, then the volume of the tetrahedron whose conterminus edges are a + b, b + c and c + a is ______.
Adjoint of ______
