Advertisements
Advertisements
प्रश्न
If A = `[(1, 0, 0),(3, 3, 0),(5, 2, -1)]`, find A−1 by the adjoint method
उत्तर
Let A = `[(1,0,0),(3,3,0),(5,2,-1)]`
∴ |A| = `|(1,0,0),(3,3,0),(5,2,-1)|`
= 1(− 3 − 0) − 0 + 0
= − 3 ≠ 0
∴ A−1 exist
A11 = (−1)1+1 M11 = `|(3,0),(2,-1)|` = −3 − 0 = −3
A12 = (−1)1+2 M12 = −`|(3,0),(5,-1)|` = −(−3 − 0) = 3
A13 = (−1)1+3 M13 = `|(3,3),(5,2)|` = 6 − 15 = −9
A21 = (−1)2+1 M21 = −`|(0,0),(2,-1)|` = −(0 − 0) = 0
A22 = (−1)2+2 M22 = `|(1, 0),(5, -1)|` = − 1 − 0 = −1
A23 = (−1)2+3 M23 = −`|(1,0),(5,2)|` = −(2 − 0) = −2
A31 =(−1)3+1 M31 = `|(0,0),(3,0)|` = 0 − 0 = 0
A32 = (−1)3+2 M32 = −`|(1,0),(3,0)|` = −(0 − 0) = 0
A33 = (−1)3+3 M33 = `|(1,0),(3,3)|` = 3 − 0 = 3
Hence, matrix of the co-factors is
`["A"_"ij"]_(3 xx 3) = [("A"_11,"A"_12,"A"_13),("A"_21,"A"_22,"A"_23),("A"_31,"A"_32,"A"_33)]`
= `[(-3,3,-9),(0,-1,-2),(0,0,3)]`
Now, adj A = `["A"_"ij"]_(3 xx 3)^"T"`
= `[(-3,0,0),(3,-1,0),(-9,-2,3)]`
∴ A−1 = `1/|"A"|` (adj A)
= `1/(-3)[(-3,0,0),(3,-1,0),(-9,-2,3)]`
= `1/3[(3,0,0),(-3,1,0),(9,2,-3)]`
APPEARS IN
संबंधित प्रश्न
Find the inverse of matrix A by using adjoint method; where A = `[(1, 0, 1), (0, 2, 3), (1, 2, 1)]`
If A = `[(1, 3), (3, 1)]`, Show that A2 - 2A is a scalar matrix.
Find the inverse of the following matrix (if they exist):
`[(2,1),(7,4)]`
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−1 = `- 1/2[(1,-4),(-1,2)]`, then A = ______.
Choose the correct alternative.
If a 3 x 3 matrix B has it inverse equal to B, thenB2 = _______
Choose the correct alternative.
If A is a 2 x 2 matrix such that A(adj. A) = `[(5, 0),(0, 5)]`, then |A| = _______
State whether the following is True or False :
If A and B are conformable for the product AB, then (AB)T = ATBT.
Find the inverse of `[(3, 1, 5),(2, 7, 8),(1, 2, 5)]` by adjoint method.
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 = `[(1, -1, 1),(2, 1, -3),(1, 1, 1)]`, 10B = `[(4, 2,2),(-5, 0, ∞),(1, -2, 3)]` and B is the inverse of matrix A, then α = ______
If A = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]`, then find the value of a31A31 + a32A32 + a33A33.
A = `[(cos theta, - sin theta),(-sin theta, -cos theta)]` then find A−1
If A is invertible matrix of order 3 and |A| = 5, then find |adj A|
If A = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`, then find A2 and hence find A−1
If A = `[(0, 1),(2, 3),(1, -1)]` and B = `[(1, 2, 1),(2, 1, 0)]`, then find (AB)−1
If A = `[(-4, -3, -3),(1, 0, 1),(4, 4, 3)]`, find adj (A).
Complete the following activity to verify A. adj (A) = det (A) I.
Given A = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` then
|A| = 2(____) – 0(____) + ( ) (____)
= 6 – 0 – 5
= ______ ≠ 0
Cofactors of all elements of matrix A are
A11 = `(-1)^2 |("( )", "( )"),("( )", "( )")|` = (______),
A12 = `(-1)^3 |(5, "( )"),("( )", 3)|` = – 15,
A13 = `(-1)^4 |(5, "( )"),("( )", 1)|` = 5,
A21 = _______, A22 = _______, A23 = _______,
A31 = `(-1)^4 |("( )", "( )"),("( )", "( )")|` = (______),
A32 = `(-1)^5 |(2, "( )"),("( )", 0)|` = ( ),
A33 = `(-1)^6 |(2, "( )"),("( )", 1)|` = 2,.
Cofactors of matrix A = `[(3, "____", "____"),("____", "____",-2),(1, "____", "____")]`
adj (A) = `[("____", "____", "____"),("____", "____","____"),("____","____","____")]`
A.adj (A) = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)] [("( )", -1, 1), (-15, "( )", -5),("( )", -2, "( )")] = [(1, 0, "( )"),("( )", "( )", "( )"),(0, "( )", "( )")]` = |A|I
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 |
If A = `[(a,b),(c,d)]` such that ad - bc ≠ 0 then A-1 is
The inverse matrix of `((3,1),(5,2))` is
If A = `[(3, -3, 4), (2, -3, 4), (0, -1, 1)]` then A-1 = ______
The inverse of `[(1,cos alpha),(- cos alpha, -1)]` is ______.
If AB = I and B = AT, then _______.
If A is a solution of x2 - 4x + 3 = 0 and `A=[[2,-1],[-1,2]],` then A-1 equals ______.
If matrix A = `[(3, -2, 4),(1, 2, -1),(0, 1, 1)]` and A–1 = `1/k` (adj A), then k is ______.
If A = `[(x, 1),(1, 0)]` and A = A–1, then x = ______.
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 the inverse of the matrix `[(α, 14, -1),(2, 3, 1),(6, 2, 3)]` does not exist, then the value of α is ______.
If A = `[(1, 1, 0),(2, 1, 5),(1, 2, 1)]`, then a11A21 + a12A22 + a13A23 is equal to ______.
If A = `[(2, 2),(-3, 2)]`, B = `[(0, -1),(1, 0)]`, then (B–1 A–1)–1 is equal to ______.
For an invertible matrix A, if A (adj A) = `|(20, 0),(0, 20)|`, then | A | = ______.
If A = `[(2, 3),(4, 5)]`, show that A2 – 7A – 2I = 0
If A = `[(1, 2, 4),(4, 3, -2),(1, 0, -3)]`. Show that A–1 exists and find A–1 using column transformation.
Find the inverse of the matrix `[(1, 1, 1),(1, 2, 3),(3, 2, 2)]` by elementary column transformation.
