Advertisements
Advertisements
प्रश्न
Find the inverse of the following matrix.
`[(1,2),(2,-1)]`
उत्तर
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
संबंधित प्रश्न
If A = `[(1, 3), (3, 1)]`, Show that A2 - 2A is a scalar matrix.
If A = `[(1,-1,2),(3,0,-2),(1,0,3)]` verify that A (adj A) = (adj A) A = | A | I
Find the inverses of the following matrices by the adjoint method:
`[(1,2,3),(0,2,4),(0,0,5)]`
Find the inverse of the following matrix (if they exist):
`((1,3),(2,7))`
Find the inverse of the following matrix (if they exist):
`[(2,-3,3),(2,2,3),(3,-2,2)]`
Choose the correct answer from the given alternatives in the following question:
If A = `[(2,-4),(3,1)]`, then the adjoint of matrix A is
Find the inverse of A = `[(1, 0, 1),(0, 2, 3),(1, 2, 1)]` by elementary column transformations.
If A = `[(1, 2),(-3, -1)], "B" = [(-1, 0),(1, 5)]`, then AB =
Check whether the following matrices are invertible or not:
`[(3, 4, 3),(1, 1, 0),(1, 4, 5)]`
If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, then A10 = ______
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 = `[("a", "b"),("c", "d")]` then find the value of |A|−1
If A = `[(0, 1),(2, 3),(1, -1)]` and B = `[(1, 2, 1),(2, 1, 0)]`, then find (AB)−1
Find the inverse of matrix B = `[(3,1, 5),(2, 7, 8),(1, 2, 5)]` by using adjoint method
If A = `[(2,3),(1,-6)]` and B = `[(-1,4),(1,-2)]`, then verify adj (AB) = (adj B)(adj A)
Find m if the matrix `[(1,1,3),(2,λ,4),(9,7,11)]` has no inverse.
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 |
The inverse matrix of `((4/5,(-5)/12),((-2)/5,1/2))` is
If A = `((-1,2),(1,-4))` then A(adj A) is
If A = `[(1,-1),(2,3)]` show that A2 - 4A + 5I2 = 0 and also find 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 = `[(4,5),(2,1)]` and A2 - 5A - 6l = 0, then A-1 = ?
If A is non-singular matrix and (A + l)(A - l) = 0 then A + A-1 = ______.
If ω is a complex cube root of unity and A = `[(ω,0,0),(0,ω^2,0),(0,0,1)]` then A-1 = ?
If A = `[(2, -3), (3, 5)]`, then |Adj A| is equal to ______
If matrix A = `[(1, -1),(2, 3)]` such that AX = I, then X is equal to ______.
If A = `[(1, 2, -1),(-1, 1, 2),(2, -1, 1)]`, then det (adj (adj A)) is ______.
If A = `[(0, 0, 1),(0, 1, 0),(1, 0, 0)]`, then A2008 is equal to ______.
Matrix A = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]` then the value of a31 A31 + a32 A32 + a33 A33 is ______.
For a invertible matrix A if A(adjA) = `[(10, 0),(0, 10)]`, then |A| = ______.
If A = `[(1, 1, 0),(2, 1, 5),(1, 2, 1)]`, then a11A21 + a12A22 + a13A23 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.
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`
if `A = [(2,-1,1),(-1,2,-1),(1,-1,2)]` then find A−1 by the adjoint method.
If A = `[(1, 2, 4),(4, 3, -2),(1, 0, -3)]`. Show that A–1 exists and find A–1 using column transformation.
