Advertisements
Advertisements
Question
Find the inverse of the following matrix.
`[(1,2),(2,-1)]`
Solution
Let A = `[(1,2),(2,-1)]`
∴ |A| = `|(1,2),(2,-1)|` = − 1 − 4 = − 5 `≠` 0
∴ A−1 exists.
consider AA−1 = I
∴ `[(1,2),(2,-1)]`A−1 = `[(1,0),(0,1)]`
By R2 − 2R1, we get
`[(1,2),(0,-5)]`A−1 = `[(1,0),(-2,1)]`
By `(-1/5)`R2, we get,
`[(1,2),(0,1)]`A−1 = `[(1,0),(2//5,-1//5)]`
By R1 − 2R2, we get,
`[(1,0),(0,1)]`A−1 = `[(1//5,2//5),(2//5,-1//5)]`
∴ A−1 = `-1/5[(-1,-2),(-2,1)]`
∴ A−1 = `1/5[(1,2),(2,-1)]`
The answer can be checked by finding the product AA−1.
AA−1 = `[(1,2),(2,-1)][(1//5,2//5),(2//5,-1//5)]`
= `[(1(1/5) + 2(2/5),1(2/5) + 2(-1/5)),(2(1/5) - 1(2/5),2(2/5) -1(-1/5))]`
= `[(1/5 + 4/5,2/5 - 2/5),(2/5 - 2/5,4/5 + 1/5)] = [(1,0),(0,1)]` = I
Hence, A−1 is the required answer.
APPEARS IN
RELATED QUESTIONS
Find the adjoint of the following matrix.
`[(2,-3),(3,5)]`
Find the adjoint of the following matrix.
`[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`
If A = `[(1,-1,2),(3,0,-2),(1,0,3)]` verify that A (adj A) = (adj A) A = | A | I
Find the inverse of the following matrix by the adjoint method.
`[(2,-2),(4,3)]`
Find the inverses of the following matrices by the adjoint method:
`[(1,2,3),(0,2,4),(0,0,5)]`
Choose the correct answer from the given alternatives in the following question:
If A = `[(lambda,1),(-1, -lambda)]`, and A-1 does not exist if λ = _______
Choose the correct answer from the given alternatives in the following question:
The inverse of A = `[(0,1,0),(1,0,0),(0,0,1)]` is
Find the inverse of the following matrices by transformation method: `[(1, 2),(2, -1)]`
If A = `[(1, 0, 1),(0, 2, 3),(1, 2, 1)] "and B" = [(1, 2, 3),(1, 1, 5),(2, 4, 7)]`, then find a matrix X such that XA = B.
Choose the correct alternative.
If a 3 x 3 matrix B has it inverse equal to B, thenB2 = _______
Fill in the blank :
If A = [aij]2x3 and B = [bij]mx1 and AB is defined, then m = _______
State whether the following is True or False :
Singleton matrix is only row matrix.
If A = `[(4, -1),(-1, "k")]` such that A2 − 6A + 7I = 0, then K = ______
If A = `[(4, 5),(2, 5)]`, then |(2A)−1| = ______
If A = `[(3, 0, 0),(0, 3, 0),(0, 0, 3)]`, then |A| |adj A| = ______
If A = `[("a", "b"),("c", "d")]` then find the value of |A|−1
If A = `[(1, 2),(3, -2),(-1, 0)]` and B = `[(1, 3, 2),(4, -1, 3)]` then find the order of AB
A + I = `[(3, -2),(4, 1)]` then find the value of (A + I)(A − I)
If A = `[(6, 5),(5, 6)]` and B = `[(11, 0),(0, 11)]` then find A'B'
If A = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`, then find A2 and hence find A−1
If A = `[(1,3,3),(1,4,3),(1,3,4)]` then verify that A(adj A) = |A| I and also find A-1.
Find the inverse of the following matrix:
`[(1,-1),(2,3)]`
Find the inverse of the following matrix:
`[(3,1),(-1,3)]`
Find m if the matrix `[(1,1,3),(2,λ,4),(9,7,11)]` has no inverse.
If X = `[(8,-1,-3),(-5,1,2),(10,-1,-4)]` and Y = `[(2,1,-1),(0,2,1),(5,p,q)]` then, find p, q if Y = X-1
A sales person Ravi has the following record of sales for the month of January, February and March 2009 for three products A, B and C. He has been paid a commission at fixed rate per unit but at varying rates for products A, B and C.
| Months | Sales in units | Commission | ||
| A | B | C | ||
| January | 9 | 10 | 2 | 800 |
| February | 15 | 5 | 4 | 900 |
| March | 6 | 10 | 3 | 850 |
Find the rate of commission payable on A, B and C per unit sold using matrix inversion method.
The prices of three commodities A, B, and C are ₹ x, y, and z per unit respectively. P purchases 4 units of C and sells 3 units of A and 5 units of B. Q purchases 3 units of B and sells 2 units of A and 1 unit of C. R purchases 1 unit of A and sells 4 units of B and 6 units of C. In the process P, Q and R earn ₹ 6,000, ₹ 5,000 and ₹ 13,000 respectively. By using the matrix inversion method, find the prices per unit of A, B, and C.
If A and B non-singular matrix then, which of the following is incorrect?
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, -3), (3, 5)]`, then |Adj A| is equal to ______
If A2 - A + I = 0, then A-1 = ______.
The inverse of the matrix A = `[(3, 0, 0),(0, 4, 0),(0, 0, 5)]` is ______.
If A = `[(cos α, sin α),(- sin α, cos α)]`, then the matrix A is ______.
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`
If A = `[(2, 3),(4, 5)]`, show that A2 – 7A – 2I = 0
Find the inverse of the matrix `[(1, 1, 1),(1, 2, 3),(3, 2, 2)]` by elementary column transformation.
