Advertisements
Advertisements
Question
Find the inverse of the following matrix by elementary row transformations if it exists. `A=[[1,2,-2],[0,-2,1],[-1,3,0]]`
Solution
`A=[[1,2,-2],[0,-2,1],[-1,3,0]]`
`therefore A=|[1,2,-2],[0,-2,1],[-1,3,0]|`
`=1|[-2,1],[3,0]|-2|[0,1],[-1,1]|-2|[0,-2],[-1,3]|`
`|A|=1(0-3)-2(0+1)-2(0-2)`
`=-3-2+4`
`=-1!=0`
`therefore A^(-1) " exist"`
We have
`A A^(-1)=I`
`[[1,2,-2],[0,-2,1],[-1,3,0]]A^(-1)=[[1,0,0],[0,1,0],[0,0,1]]`
`R_3->R_3+R_1`
`[[1,2,-2],[0,-2,1],[0,5,-2]]A^(-1)=[[1,0,0],[0,1,0],[1,0,1]]`
`R_3->R_3+2R_2`
`[[1,2,-2],[0,-2,1],[0,1,-0]]A^(-1)=[[1,0,0],[0,1,0],[1,2,1]]`
`R_2 harr R_3`
`[[1,2,-2],[0,1,0],[0,-2,1]]A^(-1)=[[1,0,0],[1,2,1],[0,1,0]]`
`R_1->R_1-2R_2 " " R3->R_3+2R_2`
`[[1,0,-2],[0,1,0],[0,0,1]]A^(-1)=[[-1,-4,-2],[1,2,1],[2,5,2]]`
`R_1->R_1+2R_3`
`[[1,0,0],[0,1,0],[0,0,1]]A^(-1)=[[3,6,2],[1,2,1],[2,5,2]]`
`A^(-1)=[[3,6,2],[1,2,1],[2,5,2]]`
APPEARS IN
RELATED QUESTIONS
The sum of three numbers is 6. If we multiply the third number by 3 and add it to the second number we get 11. By adding first and third numbers we get a number, which is double than the second number. Use this information and find a system of linear equations. Find these three numbers using matrices.
Find the inverse of the matrix `[(1 2 3),(1 1 5),(2 4 7)]` by adjoint method
Find the inverse of the following matrix by elementary row transformations if it exists.
`A = [(1, 2, -2), (0, -2, 1), (-1, 3, 0)]`
Find the adjoint of the following matrix.
`[(2,-3),(3,5)]`
Choose the correct answer from the given alternatives in the following question:
If A = `[("cos"alpha,-"sin"alpha),("sin"alpha,"cos"alpha)]`, then A-1 = _____
Choose the correct answer from the given alternatives in the following question:
The inverse of a symmetric matrix is
Find the inverse of the following matrices by the adjoint method `[(2, -2),(4, 5)]`.
Find the inverse of the following matrices by transformation method: `[(1, 2),(2, -1)]`
Find the inverse `[(1, 2, 3 ),(1, 1, 5),(2, 4, 7)]` of the elementary row tranformation.
Find matrix X, if AX = B, where A = `[(1, 2, 3),(-1, 1, 2),(1, 2, 4)] "and B" = [(1),(2),(3)]`.
Choose the correct alternative.
If AX = B, where A = `[(-1, 2),(2, -1)], "B" = [(1),(1)]`, then X = _______
Choose the correct alternative.
If a 3 x 3 matrix B has it inverse equal to B, thenB2 = _______
Fill in the blank :
If A = `[(3, -5),(2, 5)]`, then co-factor of a12 is _______
Fill in the blank :
(AT)T = _______
Fill in the blank :
If A = `[(2, 1),(1, 1)] "and" "A"^-1 = [(1, 1),(x, 2)]`, then x = _______
Check whether the following matrices are invertible or not:
`[(1, 2, 3),(2, 4, 5),(2, 4, 6)]`
Find inverse of the following matrices (if they exist) by elementary transformations :
`[(2, -3, 3),(2, 2, 3),(3, -2, 2)]`
Find inverse of the following matrices (if they exist) by elementary transformations :
`[(2, 0, -1),(5, 1, 0),(0, 1, 3)]`
If ω is a complex cube root of unity, then the matrix A = `[(1, ω^2, ω),(ω^2, ω, 1),(ω, 1, ω^2)]` is
If A = `[(4, -1),(-1, "k")]` such that A2 − 6A + 7I = 0, then K = ______
If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, then A10 = ______
If A = `[(4, 5),(2, 5)]`, then |(2A)−1| = ______
If A = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]`, then find the value of a31A31 + a32A32 + a33A33.
For an invertible matrix A, if A . (adj A) = `[(10, 0),(0, 10)]`, then find the value of |A|.
If A = `[(2, 2),(-3, 2)]` and B = `[(0, -1),(1, 0)]`, then find the matrix (B−1 A−1)−1.
If A = `[(3, 1),(5, 2)]`, and AB = BA = I, then find the matrix B
If A = `[(1, 2),(3, -2),(-1, 0)]` and B = `[(1, 3, 2),(4, -1, 3)]` then find the order of AB
If A = `[(6, 5),(5, 6)]` and B = `[(11, 0),(0, 11)]` then find A'B'
Find the adjoint of matrix A = `[(6, 5),(3, 4)]`
If A = `[(-4, -3, -3),(1, 0, 1),(4, 4, 3)]`, find adj (A).
Find the inverse of A = `[(sec theta, tan theta, 0),(tan theta, sec theta, 0),(0, 0, 1)]`
Choose the correct alternative:
If A is a non singular matrix of order 3, then |adj (A)| = ______
Find the adjoint of the matrix A = `[(2,3),(1,4)]`
Solve by matrix inversion method:
2x – z = 0; 5x + y = 4; y + 3z = 5
The sum of three numbers is 20. If we multiply the first by 2 and add the second number and subtract the third we get 23. If we multiply the first by 3 and add second and third to it, we get 46. By using the matrix inversion method find the numbers.
Weekly expenditure in an office for three weeks is given as follows. Assuming that the salary in all the three weeks of different categories of staff did not vary, calculate the salary for each type of staff, using the matrix inversion method.
| Week | Number of employees | Total weekly salary (in ₹) |
||
| A | B | C | ||
| 1st week | 4 | 2 | 3 | 4900 |
| 2nd week | 3 | 3 | 2 | 4500 |
| 3rd week | 4 | 3 | 4 | 5800 |
adj (AB) is equal to:
The inverse matrix of `((3,1),(5,2))` is
If A = `((-1,2),(1,-4))` then A(adj A) is
If A and B non-singular matrix then, which of the following is incorrect?
If A = `[(1,-1),(2,3)]` show that A2 - 4A + 5I2 = 0 and also find A-1.
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.
If A = `[(1,2),(3,-5)]`, then A-1 = ?
The matrix M = `[(0,1,2),(1,2,3),(3,1,1)]` and its inverse is N = [nij]. What is the element n23 of matrix N?
If A = `[(2,3),(1,2)]`, B = `[(1,0),(3,1)]`, then B-1A-1 = ?
If A = `[(2, 0, -1), (5, 1, 0), (0, 1, 3)]` and A−1 = `[(3, -1, 1), (α, 6, -5), (β, -2, 2)]`, then the values of α and β are, respectively.
If A is non-singular matrix such that (A - 2l)(A - 4l) = 0 then A + 8A-1 = ______.
If A = `[(p/4, 0, 0), (0, q/5, 0), (0, 0, r/6)]` and `"A"^-1 = [(1/4, 0, 0), (0, 1/5, 0), (0, 0, 1/6)]`, then p + q + r = ______
If A = `[(0, -1, 0), (1, 0, 0), (0, 0, -1)]`, then A-1 is ______
If A is non-singular matrix and (A + l)(A - l) = 0 then A + A-1 = ______.
If A = `[(1 + 2"i", "i"),(- "i", 1 - 2"i")]`, where i = `sqrt-1`, then A(adj A) = ______.
If A and Bare square matrices of order 3 such that |A| = 2, |B| = 4, then |A(adj B)| = ______.
The inverse of `[(1,cos alpha),(- cos alpha, -1)]` is ______.
If A = `[(1,-1,1),(2,1,-3),(1,1,1)]`, then the sum of the elements of A-1 is ______.
If AB = I and B = AT, then _______.
If A = `[(2, 2),(4, 5)]` and A–1 = λ(adj(A)), then λ = ______ .
If matrix A = `[(1, -1),(2, 3)]` such that AX = I, then X is equal to ______.
If matrix A = `[(3, -2, 4),(1, 2, -1),(0, 1, 1)]` and A–1 = `1/k` (adj A), then k is ______.
Matrix A = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]` then the value of a31 A31 + a32 A32 + a33 A33 is ______.
If A = `[(2, 2),(-3, 2)]`, B = `[(0, -1),(1, 0)]`, then (B–1 A–1)–1 is equal to ______.
If matrix A = `[(1, 2),(4, 3)]`, such that AX = I, then X is equal to ______.
If A = `[(cos α, sin α),(-sin α, cos α)]`, then find α satisfying `0 < α < π/2`, when A + AT = `sqrt(2) l_2` where AT is transpose of A.
For an invertible matrix A, if A (adj A) = `|(20, 0),(0, 20)|`, then | A | = ______.
If A = `[(1, 2),(3, 4)]` verify that A (adj A) = (adj A) A = |A| I
If A = `[(4, 3, 2),(-1, 2, 0)]`, B = `[(1, 2),(-1, 0),(1, -2)]`
Find (AB)–1 by adjoint method.
Solution:
AB = `[(4, 3, 2),(-1, 2, 0)] [(1, 2),(-1, 0),(1, -2)]`
AB = [ ]
|AB| = `square`
M11 = –2 ∴ A11 = (–1)1+1 . (–2) = –2
M12 = –3 A12 = (–1)1+2 . (–3) = 3
M21 = 4 A21 = (–1)2+1 . (4) = –4
M22 = 3 A22 = (–1)2+2 . (3) = 3
Cofactor Matrix [Aij] = `[(-2, 3),(-4, 3)]`
adj (A) = [ ]
A–1 = `1/|A| . adj(A)`
A–1 = `square`
Find the inverse of the matrix `[(1, 1, 1),(1, 2, 3),(3, 2, 2)]` by elementary column transformation.
