Advertisements
Advertisements
प्रश्न
Answer the following question:
If A = `[(1, omega),(omega^2, 1)]`, B = `[(omega^2, 1),(1, omega)]`, where ω is a complex cube root of unity, then show that AB + BA + A −2B is a null matrix
Advertisements
उत्तर
ω is a complex cube root of unity
∴ ω3 = 1 and ω4 = ω3·ω = ω ...(1)
Also 1 + ω + ω2 = 0 ...(2)
AB = `[(1, omega),(omega^2, 1)] [(omega^2, 1),(1, omega)]`
= `[(omega^2 + omega,1 + omega^2),(omega^4 + 1, omega^2 + omega)]`
BA = `[(omega^2, 1),(1, omega)] [(1, omega),(omega^2, 1)]`
= `[(omega^2 + omega^2, omega^3 + 1),(1 + omega^3, omega + omega)]`
= `[(2omega^2, 2),(2, 2omega)]` ...[∵ ω3 = 1]
∴ AB + BA + A – 2B
= `[(omega^2 + omega, 1 + omega^2),(omega^4 + 1, omega^2 + omega)] + [(2omega^2, 2),(2, 2omega)] + [(1, omega),(omega^2, 1)] -2[(omega^2, 1),(1, omega)]`
= `[(omega^2 + omega, 1 + omega^2),(omega^4 + 1, omega^2 + omega)] + [(2omega^2, 2),(2, 2omega)] + [(1, omega),(omega^2, 1)] - [(2omega^2, 2),(2, 2omega)]`
= `[(omega^2 + omega + 2omega^2 + 1 - 2omega^2, 1 + omega^2 + 2 + omega - 2),(omega^4 + 1 + 2 + omega^2 - 2,omega^2 + omega + 2omega + 1 - 2omega)]`
= `[(1 + omega + omega^2, 1 + omega + omega^2),(1 + omega + omega^2, 1 + omega + omega^2)]` ...[∵ ω4 = ω]
= `[(0, 0),(0, 0)]` ...[By (2)]
which is a null matrix.
APPEARS IN
संबंधित प्रश्न
If for any 2 x 2 square matrix A, `A("adj" "A") = [(8,0), (0,8)]`, then write the value of |A|
Find the value of x, y, and z from the following equation:
`[(4,3),(x,5)] = [(y,z),(1,5)]`
Let A = `[(0,1),(0,0)]`show that (aI+bA)n = anI + nan-1 bA , where I is the identity matrix of order 2 and n ∈ N
if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = `[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))]` `n in N`
Use product `[(1,-1,2),(0,2,-3),(3,-2,4)][(-2,0,1),(9,2,-3),(6,1,-2)]` to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3
A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?
Using coding matrix A=`[(2,1),(3,1)]` encode the message THE CROW FLIES AT MIDNIGHT.
If A = `[[0 , 2],[3, -4]]` and kA = `[[0 , 3"a"],[2"b", 24]]` then find the value of k,a and b.
If A and B are square matrices of the same order 3, such that ∣A∣ = 2 and AB = 2I, write the value of ∣B∣.
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[9 sqrt(2) -3]`
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[(2, 0, 0),(3, -1, 0),(-7, 3, 1)]`
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[(0, 0, 1),(0, 1, 0),(1, 0, 0)]`
Identify the following matrix is singular or non-singular?
`[(5, 0, 5),(1, 99, 100),(6, 99, 105)]`
Find k if the following matrix is singular:
`[(7, 3),(-2, "k")]`
If A = `[(5, 1, -1),(3, 2, 0)]`, Find (AT)T.
If A = `[(7, 3, 1),(-2, -4, 1),(5, 9, 1)]`, Find (AT)T.
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`
If A = `[(6, 0),("p", "q")]` is a scalar matrix, then the values of p and q are ______ respectively.
If two matrices A and B are of the same order, then 2A + B = B + 2A.
For the non singular matrix A, (A′)–1 = (A–1)′.
AB = AC ⇒ B = C for any three matrices of same order.
If A = `[(3, -4),(1, 1),(2, 0)]` and B = `[(2, 1, 2),(1, 2, 4)]`, then verify (BA)2 ≠ B2A2
Show by an example that for A ≠ O, B ≠ O, AB = O
If `[("a","b"),("c", "-a")]`is a square root of the 2 x 2 identity matrix, then a, b, c satisfy the relation ____________.
The matrix `[(0,5,-7),(-5,0,11),(7,-11,0)]` is ____________.
If `[(1,2),(3,4)],` then A2 - 5A is equal to ____________.
`root(3)(4663) + 349` = ? ÷ 21.003
`[(5sqrt(7) + sqrt(7)) + (4sqrt(7) + 8sqrt(7))] - (19)^2` = ?
A matrix is said to be a column matrix if it has
A matrix is said to be a row matrix, if it has
A diagonal matrix is said to be a scalar matrix if its diagonal elements are
If all the elements are zero, then matrix is said to be
The number of all possible matrices of order 3/3, with each entry 0 or 1 is
Find X, If `[X - 5 - 1] [(1, 0, 2),(0, 2, 1),(2, 0, 3)][(x),(4),(1)] ` = 0
If A and B are square matrices of order 3 × 3 and |A| = –1, |B| = 3, then |3AB| equals ______.
If A is a square matrix of order 3, then |2A| is equal to ______.
Assertion: Let the matrices A = `((-3, 2),(-5, 4))` and B = `((4, -2),(5, -3))` be such that A100B = BA100
Reason: AB = BA implies AB = BA for all positive integers n.
