Advertisements
Advertisements
प्रश्न
A = `[(cos theta, - sin theta),(-sin theta, -cos theta)]` then find A−1
उत्तर
|A| = `|(cos theta, - sin theta),(-sin theta, - cos theta)|`
= – cos2θ – sin2θ
= –1 ≠ 0
∴ A–1 exists.
A11 = (–1)1+1 M11 = M11 = – cos θ
A12 = (–1)1+2 M12 = – M12 = sin θ
A21 = (–1)2+1 M21 = – M21 = sin θ
A22 = (–1)2+2 M22 = M22 = cos θ
∴ adj (A) = `[(- cos theta, sin theta),(sin theta, cos theta)]^"T"`
= `[(- cos theta, sin theta),(sin theta, cos theta)]`
A−1 = `1/|"A"|` adj (A)
= `[(cos theta, -sin theta),(-sin theta, -cos theta)]`
APPEARS IN
संबंधित प्रश्न
Find the matrix of the co-factor for the following matrix.
`[(1,3),(4,-1)]`
Find the adjoint of the following matrix.
`[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`
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:
For a 2 × 2 matrix A, if A(adj A) = `[(10,0),(0,10)]`, then determinant A equals
Find the inverse of the following matrices by the adjoint method `[(1, 2, 3),(0, 2, 4),(0, 0, 5)]`.
Find the inverse of the following matrices by transformation method: `[(1, 2),(2, -1)]`
Find the inverse of the following matrices by transformation method:
`[(2, 0, −1),(5, 1, 0),(0, 1, 3)]`
Find the inverse `[(1, 2, 3 ),(1, 1, 5),(2, 4, 7)]` of the elementary row tranformation.
State whether the following is True or False :
Singleton matrix is only row matrix.
Solve the following :
If A = `[(2, -3),(3, -2),(-1, 4)],"B" = [(-3, 4, 1),(2, -1, -3)]`, verify (3A – 5BT)T = 3AT – 5B.
Check whether the following matrices are invertible or not:
`[(1, 0),(0, 1)]`
Find inverse of the following matrices (if they exist) by elementary transformations :
`[(1, -1),(2, 3)]`
If `[(x - y - z),(-y + z),(z)] = [(0),(5),(3)]`, then the value of x, y and z are respectively ______
If A = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]`, then find the value of a31A31 + a32A32 + a33A33.
If f(x) = x2 − 2x − 3 then find f(A) when A = `[(1, 2),(2, 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).
If A = [aij]2×2, where aij = i – j, then A = ______
Complete the following activity to find inverse of matrix using elementary column transformations and hence verify.
`[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` B−1 = `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
C1 → C1 + C3
`[("( )", 0, -1),("( )", 1, 0),("( )", 1, 3)]` B−1 = `[("( )", 0, 0),("( )", 1, 0),("( )", 0, 1)]`
C3 → C3 + C1
`[(1, 0, 0),("( )", 1, "( )"),(3, 1, "( )")]` B−1 = `[(1, 0, "( )"),(0, 1, 0),("( )", 0, "( )")]`
C1 → C1 – 5C2, C3 → C3 – 5C2
`[(1, "( )", 0),(0, 1, 0),("( )", 1, "( )")]` B−1 = `[(1, 0, "( )"),("( )", 1, -5),(1, "( )", 2)]`
C1 → C1 – 2C3, C2 → C2 – C3
`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]` B−1 = `[(3, -1, "( )"),("( )", 6, -5),(5, "( )", "( )")]`
B−1 = `[("( )", "( )", "( )"),("( )", "( )", "( )"),("( )", "( )", "( )")]`
`[(2, "( )", -1),("( )", 1, 0),(0, 1, "( )")] [(3, "( )", "( )"),("( )", 6, "( )"),("( )", -2, "( )")] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
Find the inverse of the following matrix:
`[(-3,-5,4),(-2,3,-1),(1,-4,-6)]`
If A-1 = `[(1,0,3),(2,1,-1),(1,-1,1)]` then, find A.
Show that the matrices A = `[(2,2,1),(1,3,1),(1,2,2)]` and B = `[(4/5,(-2)/5,(-1)/5),((-1)/5,3/5,(-1)/5),((-1)/5,(-2)/5,4/5)]` are inverses of each other.
Solve by matrix inversion method:
x – y + 2z = 3; 2x + z = 1; 3x + 2y + z = 4
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.
If A and B non-singular matrix then, which of the following is incorrect?
If A = `|(1,1,1),(3,4,7),(1,-1,1)|` verify that A(adj A) = (adj A)(A) = |A|I3.
If A = `[(2,3),(1,2)]`, B = `[(1,0),(3,1)]`, then B-1A-1 = ?
The sum of the cofactors of the elements of second row of the matrix `[(1, 3, 2), (-2, 0, 1), (5, 2, 1)]` is ____________.
If A = `[(0, -1, 0), (1, 0, 0), (0, 0, -1)]`, then A-1 is ______
If A = `[(2, -3), (3, 5)]`, then |Adj A| is equal to ______
If A = `[(cos theta, sin theta, 0),(-sintheta, costheta, 0),(0, 0, 1)]`, where A11, A11, A13 are co-factors of a11, a12, a13 respectively, then the value of a11A11 + a12A12 + a13A13 = ______.
The inverse of the matrix A = `[(3, 0, 0),(0, 4, 0),(0, 0, 5)]` is ______.
If A = `[(2, -3, 3),(2, 2, 3),(3, "p", 2)]` and A–1 = `[(-2/5, 0, 3/5),(-1/5, 1/5, "q"),(2/5, 1/5, -2/5)]`, then ______.
A–1 exists if |A| = 0.
For a invertible matrix A if A(adjA) = `[(10, 0),(0, 10)]`, then |A| = ______.
if `A = [(2,-1,1),(-1,2,-1),(1,-1,2)]` then find A−1 by the adjoint method.
If A = `[(2, 3),(4, 5)]`, show that A2 – 7A – 2I = 0
