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 = `[(0, -tan α/2), (tan α/2, 0)]` and I is the identity matrix of order 2, show that I + A = `(I - A)[(cos α, -sin α),(sin α, cos α)]`
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`
if `A = [(3,-4),(1,-1)]` then prove A"=` [(1+2n, -4n),(n, 1-2n)]` where n is any positive integer
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 the matrix A =`[(0,a,-3),(2,0,-1),(b,1,0)]` is skew symmetric, Find the value of 'a' and 'b'
If 𝒙 = r cos θ and y= r sin θ prove that JJ-1=1.
If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.
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, 4, 7),(-4, 0, -3),(-7, 3, 0)]`
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:
`[(6, 0),(0, 6)]`
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?
`[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]`
Identify the following matrix is singular or non-singular?
`[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]`
Find k if the following matrix is singular:
`[(7, 3),(-2, "k")]`
Find k if the following matrix is singular:
`[(4, 3, 1),(7, "k", 1),(10, 9, 1)]`
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.
Select the correct option from the given alternatives:
Given A = `[(1, 3),(2, 2)]`, I = `[(1, 0),(0, 1)]` if A – λI is a singular matrix then _______
Answer the following question:
If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find 2A + B – 5C
If A = `[(6, 0),("p", "q")]` is a scalar matrix, then the values of p and q are ______ respectively.
Choose the correct alternative:
If A = `[(2, 0),(0, 2)]`, then A2 – 3I = ______
State whether the following statement is True or False:
If A is non singular, then |A| = 0
If A = `[(2, 0, 0),(0, 1, 0),(0, 0, 1)]`, then |adj (A)| = ______
Show by an example that for A ≠ O, B ≠ O, AB = O
A square matrix A = [aij]nxn is called a diagonal matrix if aij = 0 for ____________.
If A is a square matrix, then A – A’ is a ____________.
If A `= [("cos x", - "sin x"),("sin x", "cos x")]`, find AAT.
If the matrix A `= [(5,2,"x"),("y",2,-3),(4, "t",-7)]` is a symmetric matrix, then find the value of x, y and t respectively.
The matrix `[(0,5,-7),(-5,0,11),(7,-11,0)]` is ____________.
`root(3)(4663) + 349` = ? ÷ 21.003
A = `[a_(ij)]_(m xx n)` is a square matrix, if
If the sides a, b, c of ΔABC satisfy the equation 4x3 – 24x2 + 47x – 30 = 0 and `|(a^2, (s - a)^2, (s - a)^2),((s - b)^2, b^2, (s - b)^2),((s - c)^2, (s - c)^2, c^2)| = p^2/q` where p and q are co-prime and s is semiperimeter of ΔABC, then the value of (p – q) is ______.
How many matrices can be obtained by using one or more numbers from four given numbers?
If A = `[(5, x),(y, 0)]` and A = AT, where AT is the transpose of the matrix A, then ______.
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 ______.
