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 A is a square matrix, such that A2=A, then write the value of 7A−(I+A)3, where I is an identity matrix.
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:
`[(x + y, 2),(5 + z, xy)] = [(6, 2), (5, 8)]`
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 and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N
If A = `[(α, β),(γ, -α)]` is such that A2 = I, then ______.
If A is a square matrix such that A2 = A, then (I + A)3 – 7A is equal to ______.
In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix. Using scalar multiplication, find the total number of posts of each kind in all the colleges.
Using coding matrix A=`[(2,1),(3,1)]` encode the message THE CROW FLIES AT MIDNIGHT.
If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.
if `vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`
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:
`[(5),(4),(-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:
`[(10, -15, 27),(-15, 0, sqrt(34)),(27, sqrt(34), 5/3)]`
Find k if the following matrix is singular:
`[(7, 3),(-2, "k")]`
Find k if the following matrix is singular:
`[("k" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]`
If A = `[(1, 0),(-1, 7)]`, find k so that A2 – 8A – kI = O, where I is a unit matrix and O is a null matrix of order 2.
If A = `[(3, 1),(-1, 2)]`, prove that A2 – 5A + 7I = 0, where I is unit matrix of order 2
Choose the correct alternative:
If B = `[(6, 3),(-2, "k")]` is singular matrix, then the value of k is ______
If A is a square matrix of order 2 such that A(adj A) = `[(7, 0),(0, 7)]`, then |A| = ______
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
Given A = `[(2, 4, 0),(3, 9, 6)]` and B = `[(1, 4),(2, 8),(1, 3)]` is (AB)′ = B′A′?
If a matrix A is both symmetric and skew-symmetric, then ____________.
The matrix `[(0,-5,8),(5,0,12),(-8,-12,0)]` is a ____________.
`root(3)(4663) + 349` = ? ÷ 21.003
`[(5sqrt(7) + sqrt(7)) + (4sqrt(7) + 8sqrt(7))] - (19)^2` = ?
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.
A matrix which is both symmetric and skew symmetric matrix is a ______.
What is the order of \[A=\begin{bmatrix}0\\ \sqrt{3}\\ -1\\ \dfrac{1}{2}\end{bmatrix}\]?
What is the order of \[B=\begin{bmatrix}-\dfrac{1}{2}&\sqrt{5}&2&3\end{bmatrix}\]?
Which general form represents a square matrix?
A scalar matrix is a diagonal matrix in which:
Which matrix is a scalar matrix?
A zero matrix (null matrix) is a matrix in which:
