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 = I, then find the simplified value of (A – I)3 + (A + I)3 – 7A.
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)]`
Find the value of x, y, and z from the following equation:
`[(x + y + z), (x + z), (y + z)] = [(9), (5), (7)]`
`A = [a_(ij)]_(m xx n)` is a square matrix, if ______.
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
Investigate for what values of λ and μ the equations
2x + 3y + 5z = 9
7x + 3y - 2z = 8
2x + 3y + λz = μ have
A. No solutions
B. Unique solutions
C. An infinite number of solutions.
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:
`[(3, -2, 4),(0, 0, -5),(0, 0, 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:
`[(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:
`[(2, 0, 0),(3, -1, 0),(-7, 3, 1)]`
Identify the following matrix is singular or non-singular?
`[(5, 0, 5),(1, 99, 100),(6, 99, 105)]`
Identify the following matrix is singular or non-singular?
`[(7, 5),(-4, 7)]`
Find k if the following matrix is singular:
`[(4, 3, 1),(7, "k", 1),(10, 9, 1)]`
Construct the matrix A = [aij]3 × 3 where aij = i − j. State whether A is symmetric or skew-symmetric.
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 B = `[(6, 3),(-2, "k")]` is singular matrix, then the value of k is ______
State whether the following statement is True or False:
If A and B are two square matrices such that AB = BA, then (A – B)2 = A2 – 2AB + B2
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
2X + 3Y = `[(2, 3),(4, 0)]`, 3Y + 2Y = `[(-2, 2),(1, -5)]`
If A is a square matrix, then A – A’ is a ____________.
For any square matrix A, AAT 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.
If a matrix A is both symmetric and skew-symmetric, then ____________.
`root(3)(4663) + 349` = ? ÷ 21.003
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 ______.
If D = `[(0, aα^2, aβ^2),(bα + c, 0, aγ^2),(bβ + c, (bγ + c), 0)]` is a skew-symmetric matrix (where α, β, γ are distinct) and the value of `|((a + 1)^2, (1 - a), (2 - c)),((3 + c), (b + 2)^2, (b + 1)^2),((3 - b)^2, b^2, (c + 3))|` is λ then the value of |10λ| is ______.
If A and B are square matrices of order 3 × 3 and |A| = –1, |B| = 3, then |3AB| equals ______.
If `A = [(1,-1,2),(0,-1,3)], B = [(-2,1),(3,-1),(0,2)],` then AB is a singular matrix.
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.
