Advertisements
Advertisements
Question
Solve the following equations by the method of inversion:
2x + 3y = - 5, 3x + y = 3
Advertisements
Solution
2x + 3y = - 5
3x + y = 3
The matrix form of above equation is
AX = B,
`"A" = [(2,3),(3,1)], "X" = [("x"),("y")] "and" "B" = [(-5),(3)]`
|A| = `|(2,3),(3,1)|` = 2 - 9 = -7 ≠ 0
A-1 exists.
adj A = `[(1,-3),(-3,2)]`
`"A"^-1 = 1/|"A"|` adj A
∴ `"A"^-1 = 1/-7[(1,-3),(-3,2)]`
Now, premultiply AX = B by `"A"^-1`, we get,
A-1(AX) = A-1B
∴ (A-1A)X = A-1B
∴ IX = A-1B
`[("x"),("y")] = (-1)/7 [(-5-9),(15+6)]`
∴ `[("x"),("y")] = (-1)/7 [(-14),(21)]`
∴ `[("x"),("y")] = [(2),(-3)]`
by equality of matrix
∴ x = 2, y = - 3
APPEARS IN
RELATED QUESTIONS
Solve the following equations by inversion method.
x + 2y = 2, 2x + 3y = 3
Solve the following equations by inversion method.
2x + 6y = 8, x + 3y = 5
Solve the following equations by the reduction method.
2x + y = 5, 3x + 5y = – 3
Solve the following equations by the reduction method.
3x – y = 1, 4x + y = 6
Solve the following equations by inversion method:
x + y = 4, 2x – y = 5
Solve the following equation by the method of inversion:
2x - y = - 2, 3x + 4y = 3
Solve the following equations by the method of inversion:
x + y+ z = 1, 2x + 3y + 2z = 2,
ax + ay + 2az = 4, a ≠ 0.
Solve the following equation by the method of inversion:
5x − y + 4z = 5, 2x + 3y + 5z = 2 and 5x − 2y + 6z = −1
Express the following equations in matrix form and solve them by the method of reduction:
x − y + z = 1, 2x − y = 1, 3x + 3y − 4z = 2
Express the following equations in matrix form and solve them by the method of reduction:
`x + y = 1, y + z = 5/3, z + x 4/33`.
Express the following equations in matrix form and solve them by the method of reduction:
2x - y + z = 1, x + 2y + 3z = 8, 3x + y - 4z = 1.
Express the following equations in matrix form and solve them by the method of reduction:
x + 2y + z = 8, 2x + 3y – z = 11, 3x – y – 2z = 5.
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.
The sum of three numbers is 6. Thrice the third number when added to the first number, gives 7. On adding three times the first number to the sum of second and third numbers, we get 12. Find the three number by using matrices.
Express the following equations in matrix form and solve them by the method of reduction:
x + 3y + 2z = 6,
3x − 2y + 5z = 5,
2x − 3y + 6z = 7
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
Solve the following equation by the method of inversion.
2x – y + z = 1,
x + 2y + 3z = 8,
3x + y – 4z = 1
Solve the following equations by method of inversion:
x + y + z = 1, x – y + z = 2 and x + y – z = 3
Express the following equations in matrix form and solve them by method of reduction.
x + 3y = 2, 3x + 5y = 4
Express the following equations in matrix form and solve them by the method of reduction.
x + y + z = 1, 2x + 3y + 2z = 2 and x + y + 2z = 4
If x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6, then (y, z) = _______
Find x, y, z, if `{5[(0, 1),(1, 0),(1, 1)] - [(2, 1),(3, - 2),(1, 3)]} [(2),(1)] = [(x - 1),(y + 1),(2z)]`
Solve the following equations by method of inversion : x + y – z = 2, x – 2y + z = 3 and 2x – y – 3z = – 1
Solve the following equations by method of reduction:
x + 2y + z = 3, 3x – y + 2z = 1 and 2x – 3y + 3z = 2
Solve the following equations by method of reduction :
x – 3y + z = 2 , 3x + y + z = 1 and 5x + y + 3z = 3
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.
Complete the following activity.
The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.
Solution: Let the cost of potato, wheat and rice per kg be x, y and z respectively.
Therefore by given conditions,
4x + ( )y + 2( ) = ( )
x + 2y + ( )( ) = ( )
( )x + 2y + 3z = ( )
Matrix form of above equations is,
`[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]`
R1 ↔ R2
`[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]`
R2 – 4R1, R3 – 6R1
`[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]`
`(-1)/5 "R"_2, (-1)/5 "R"_3`
`[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]`
R3 – 2R2
`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]`
By pre multiplying we get,
x + 2y + ( )z = ( ) .....(i)
y + 2z = 24 ......(ii)
–z = ( ) ......(iii)
From (iii), we get, z = ( )
From (ii), we get, y = ( )
From (i), we get, x = ( )
Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.
If `[(1, -1, x), (1, x, 1), (x, -1, 1)]` has no inverse, then the real value of x is ______
If A =`[(1, -1), (2, 3)]` and adj (A) = `[(a, b), (-2, 1)]`, then ______
Solve the following system of equations by the method of inversion.
x – y + z = 4, 2x + y – 3z = 0, x + y + z = 2
