Advertisements
Advertisements
Question
Answer the following question:
If A = `[(costheta, sintheta),(-sintheta, costheta)]`, prove that An = `[(cos"n"theta, sin"n"theta),(-sin"n"theta, cos"n"theta)]`, for all n ∈ N
Solution
A = `[(costheta, sintheta),(-sintheta, costheta)]`
Let P(n) ≡ An = `[(cos"n"theta, sin"n"theta),(-sin"n"theta, cos"n"theta)]`, for all n ∈ N.
Put n = 1
∴ R.H.S. = `[(costheta, sintheta),(-sintheta, costheta)]` = A = L.H.S.
∴ P(n) is true for n = 1.
Let us consider that P(n) is true for n = k.
∴ AK = `[(cos"k"theta, sin"k"theta),(-sin"k"theta, cos"k"theta)]` ...(i)
We have to prove that P(n) is true for
n = k + 1,
i.e., to prove that
Ak+1 = `[(cos("k" + 1)theta , sin("k" + 1)theta),(-sin("k" + 1)theta, cos("k" + 1)theta)]`
R.H.S. = `[(cos("k" + 1)theta , sin("k" + 1)theta),(-sin("k" + 1)theta, cos("k" + 1)theta)]`
L.H.S. = Ak+1 = Ak.A
= `[(cos"k"theta, sin"k"theta),(-sin"k"theta, cos"k"theta)] [(costheta, sintheta),(-sintheta, costheta)]` ...[From (i)]
= `[(cos"k"theta costheta - sin"k"theta sintheta, cos"k"theta sintheta + sin"k"theta costheta),(-sin"k"theta costheta - cos"k"theta sintheta, -sin"k"theta sintheta + cos"k"theta costheta)]`
= `[(cos"k"theta costheta - sin"k"theta sintheta, sin"k" theta costheta + cos"k"theta sintheta),(-(sin"k"theta costheta + cos"k"theta sintheta), cos"k"theta costheta - sin"k"theta sintheta)]`
= `[(cos("k"theta + theta), sin("k"theta + theta)),(-sin("k"theta + theta), cos("k"theta + theta))]`
= `[(cos("k" + 1)theta, sin("k" + 1)theta),(-sin("k" + 1)theta, cos("k" + 1)theta)]`
= R.H.S.
∴ P(n) is true for n = k + 1.
From all steps above, by the principle of Mathematical induction, P(n) is true for all n ∈ N.
∴ An = `[(cos"n"theta, sin"n"theta),(-sin"n"theta, cos"n"theta)]` for all n ∈ N.
APPEARS IN
RELATED QUESTIONS
Find AT, if A = `[(2, -6, 1),(-4, 0, 5)]`
If [aij]3×3 where aij = 2(i – j). Find A and AT. State whether A and AT are symmetric or skew-symmetric matrices?
If A = `[(5, -3),(4, -3),(-2, 1)]`, Prove that (2A)T = 2AT
If A = `[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]` where i = `sqrt(-1)` Prove that AT = – A
If A = `[(2, -3),(5, -4),(-6, 1)]`, B = `[(2, 1),(4, -1),(-3, 3)]` and C = `[(1, 2),(-1, 4),(-2, 3)]` then show that (A + B)T = AT + BT
If A = `[(2, -3),(5, -4),(-6, 1)]`, B = `[(2, 1),(4, -1),(-3, 3)]` and C = `[(1, 2),(-1, 4),(-2, 3)]` then show that (A – C)T = AT – CT
If A = `[(5, 4),(-2, 3)]` and B = `[(-1, 3),(4, -1)]`, then find CT , such that 3A – 2B + C = I, where I is the unit matrix of order 2
If A = `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find AT + 4BT
If A = `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find 5AT – 5BT
If A = `[(1, 0, 1),(3, 1, 2)]`, B = `[(2, 1, -4),(3, 5, -2)]` and C = `[(0, 2, 3),(-1, -1, 0)]`, verify that (A + 2B + 3C)T = AT + 2BT + 3CT
If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + BT)T = AT + B
Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where
A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`
Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where
A = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)]`
Express the following matrix as the sum of a symmetric and a skew symmetric matrix
`[(4, -2),(3, -5)]`
Express the following matrix as the sum of a symmetric and a skew symmetric matrix
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
If A = `[(2, -1),(3, -2),(4, 1)]` and B = `[(0, 3, -4),(2, -1, 1)]`, verify that (BA)T = AT BT
If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, show that ATA = I, where I is the unit matrix of order 2
Select the correct option from the given alternatives:
Consider the matrices A = `[(4, 6, -1),(3, 0, 2),(1, -2, 5)]`, B = `[(2, 4),(0, 1),(-1, 2)]`, C = `[(3),(1),(2)]` out of the given matrix product ________
i) (AB)TC
ii) CTC(AB)T
iii) CTAB
iv) ATABBTC
Select the correct option from the given alternatives:
If A = `[(1, 2, 2),(2, 1, -2),("a", 2, "b")]` is a matrix satisfying the equation AAT = 9I, where I is the identity matrix of order 3, then the ordered pair (a, b) is equal to ________
Select the correct option from the given alternatives:
If A = `[(alpha, 2),(2, alpha)]` and |A3| = 125, then α = _______
Select the correct option from the given alternatives:
For suitable matrices A, B, the false statement is _____
Answer the following question:
If A = `[(2, -3),(3, -2),(-1, 4)]`, B = `[(-3, 4, 1),(2, -1, -3)]` Verify (A + 2BT)T = AT + 2B
Answer the following question:
If A = `[(cosalpha, -sinalpha),(sinalpha, cosalpha)]` and A + AT = I, where I is unit matrix 2 × 2, then find the value of α
Answer the following question:
If A = `[(2, 1, -3),(0, 2, 6)]`, B = `[(1, 0, -2),(3, -1, 4)]`, find ABT and ATB
Answer the following question:
If A = `[(2, -4),(3, -2),(0, 1)]`, B = `[(1, -1, 2),(-2, 1, 0)]`, show that (AB)T = BTAT
Answer the following question:
If A = `[(3, -4),(1, -1)]`, prove that An = `[(1 + 2"n", -4"n"),("n", 1 - 2"n")]`, for all n ∈ N
