Advertisements
Advertisements
प्रश्न
Find the inverse of A = `[("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)]` by elementary row transformations.
Advertisements
उत्तर
|A| = `|("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)|`
= cos θ(cos θ – 0) + sin θ(sin θ – 0) + 0
= cos2θ + sin2θ
= 1 ≠ 0
∴ A-1 exists.
Consider AA–1 = I
∴ `[("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)] "A"^-1 = [(1,0,0),(0,1,0),(0,0,1)]`
By cos θ × R1, we get,
`[("cos"^2theta, -"sin"theta "cos"theta,0),("sin"theta, "cos"theta, 0),(0,0,1)] "A"^-1 = [("cos"theta,0,0),(0,1,0),(0,0,1)]`
By R1 + sin θ × R2, we get,
`[(1,0,0),("sin"theta,"cos"theta,0),(0,0,1)] "A"^-1 = [("cos"theta,"sin"theta,0),(0,1,0),(0,0,1)]`
By R2 – sin θ × R1, we get,
`[(1,0,0),(0,"cos"theta,0),(0,0,1)] "A"^-1 = [("cos"theta,"sin"theta, 0),(-"sin"theta"cos"theta,"cos"^2theta,0),(0,0,1)]`
By `(1/("cos"theta)) xx "R"_2`, we get,
`[(1,0,0),(0,1,0),(0,0,1)] "A"^-1 = [("cos"theta,"sin"theta,0),(-"sin"theta,"cos"theta,0),(0,0,1)]`
`∴ "A"^-1 = [("cos"theta,"sin"theta,0),(-"sin"theta,"cos"theta,0),(0,0,1)]`
APPEARS IN
संबंधित प्रश्न
Apply the given elementary transformation of the following matrix.
A = `[(5,4),(1,3)]`, C1↔ C2; B = `[(3,1),(4,5)]` R1↔ R2.
What do you observe?
Apply the given elementary transformation of the following matrix.
A = `[(1,2,-1),(0,1,3)]`, 2C2
B = `[(1,0,2),(2,4,5)]`, −3R1
Find the addition of the two new matrices.
Apply the given elementary transformation of the following matrix.
A = `[(1,-1,3),(2,1,0),(3,3,1)]`, 3R3 and then C3 + 2C2
Apply the given elementary transformation of the following matrix.
Use suitable transformation on `[(1,2),(3,4)]` to convert it into an upper triangular matrix.
Apply the given elementary transformation of the following matrix.
Convert `[(1,-1),(2,3)]` into an identity matrix by suitable row transformations.
Apply the given elementary transformation of the following matrix.
Transform `[(1,-1,2),(2,1,3),(3,2,4)]` into an upper triangular matrix by suitable column transformations.
If A = `[(2,1,3),(1,0,1),(1,1,1)]`, then reduce it to I3 by using row transformations.
Check whether the following matrix is invertible or not:
`[(1,0),(0,1)]`
Check whether the following matrix is invertible or not:
`((1,1),(1,1))`
Check whether the following matrix is invertible or not:
`((1,2),(3,3))`
Check whether the following matrix is invertible or not:
`((2,3),(10,15))`
Check whether the following matrix is invertible or not:
`(("sec" theta , "tan" theta),("tan" theta,"sec" theta))`
Check whether the following matrix is invertible or not:
`[(3,4,3),(1,1,0),(1,4,5)]`
Check whether the following matrix is invertible or not:
`((1,2,3),(2,-1,3),(1,2,3))`
Find the inverse of A = `[("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)]` by elementary column transformations.
If A = `[(4,5),(2,1)]`, show that `"A"^-1 = 1/6("A" - 5"I")`.
Find X, if AX = B, where A = `[(1,2,3),(-1,1,2),(1,2,4)]` and B = `[(1),(2),(3)]`
If A = `[(1,1),(1,2)], "B" = [(4,1),(3,1)]` and C = `[(24,7),(31,9)]`, then find the matrix X such that AXB = C
Find the inverse of A = `[(1,0,1),(0,2,3),(1,2,1)]` by using elementary column transformations.
Show with the usual notation that for any matrix A = `["a"_"ij"]_(3xx3) "is" "a"_11"A"_11 + "a"_12"A"_12 + "a"_13"A"_13 = |"A"|`
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 answer from the given alternatives in the following question:
The inverse of `[(0,1),(1,0)]` is
Choose the correct answer from the given alternatives in the following question:
If A = `[(1,2),(2,1)]` and A(adj A) = k I, then the value of k is
The element of second row and third column in the inverse of `[(1, 2, 1),(2, 1, 0),(-1, 0, 1)]` is ______.
If A = `[(2, -1, 1),(-2, 3, -2),(-4, 4, -3)]` the find A2
If A = `[(-2, 4),(-1, 2)]` then find A2
Find A−1 using column transformations:
A = `[(5, 3),(3, -2)]`
If A = `[(1, 2, -1),(3, -2, 5)]`, apply R1 ↔ R2 and then C1 → C1 + 2C3 on A
Find the matrix X such that `[(1, 2, 3),(2, 3, 2),(1, 2, 2)]` X = `[(2, 2, -5),(-2, -1, 4),(1, 0, -1)]`
Find the inverse of A = `[(2, -3, 3),(2, 2, 3),(3, -2, 2)]` by using elementary row transformations.
If A = `[(2, 3),(1, 2)]`, B = `[(1, 0),(3, 1)]`, find AB and (AB)−1
If A = `[(3, -1),(4, 2)]`, B = `[(2),(-1)]`, find X such that AX = B.
If A = `[(cosθ, -sinθ, 0),(sinθ, cosθ, 0),(0, 0, 1)]`, find A–1
