Advertisements
Advertisements
प्रश्न
Determine the matrices A and B if they satisfy 2A – B + `[(6, - 6, 0),(- 4, 2, 1)]` = 0 and A – 2B = `[(3, 2, 8),(-2, 1, -7)]`
उत्तर
2A – B + `[(6, - 6, 0),(- 4, 2, 1)]` = 0
A – 2B = `[(3, 2, 8),(-2, 1, -7)]`
2A – B = `- [(6, - 6, 0),(- 4, 2, 1)]` ......(1)
A – 2B = `[(3, 2, 8),(-2, 1, -7)]` .......(2)
(1) ⇒ 2A – B + `[(6, - 6, 0),(- 4, 2, 1)]`
(2) × 2 2A – 4B = `[(3, 2, 8),(-2, 1, -7)]`
Substuting 0 + 3B = `[(6, - 6, 0),(- 4, 2, 1)] - 2[(3, 2, 8),(-2, 1, -7)]`
3B = `[(- 6, 6, 0),(4, - 2, -1)] - [(6, 4, 16),(-4, 2, -14)]`
= `[(-6 - 6, 6 - 4, 0 - 16),(4 + 4, - 2 - 2, - 1 + 14)]`
3B = `[(- 12, 2, - 16),(8, -4, 13)]`
B = `1/3[(- 12, 2, - 16),(8, -4, 13)]`
Substituting for B in equation (1)
(1) ⇒ `2"A" - 1/3 [(- 12, 2, - 6),(8, -4, 13)] = - [(6, 6, 0),(- 4, 2, 1)]`
2A = `1/3[(- 1, 2, 16),(8,-4, 13)] - [(6, -6, 0),(-4, 2, 1)]`
= `[((-12)/3, 2/3, (-16)/3),(8/3, (-4)/3, 13/3)] - [(6, -6, 0),(-4, 2, 1)]`
= `[(-4 - 6, 2/3 + 6, (- 16)/3 - 0),(8/3 + 4, (-4)/3 - 2, 13/3 - 1)]`
= `[(- 10, (2 + 8)/3, (-16)/3),((8 + 12)/3, (-4 - 6)/3, (13 - 3)/3)]`
2A = `[(- 10, 20/3, (-16)/3),(20/3, (- 10)/3, 10/3)]`
A = `1/2[(- 10, 0/3, (- 16)/3),(20/3, (- 10)/3, 10/3)]`
A = `[(-5, 10/3, (-8)/3),(10/3, (-5)/3, 5/3)]`
Reqired values are
A = `[(-5, 10/3, (-8)/3),(10/3, (-5)/3, 5/3)]`
B = `1/3[(- 12, 2, - 16),(8, -4, 13)]`
APPEARS IN
संबंधित प्रश्न
Construct a 3 × 3 matrix whose elements are given by aij = `("i" + "j")^3/3`
If A = `[(1, 9),(3, 4),(8, -3)]`, B = `[(5, 7),(3, 3),(1, 0)]` then verify that A + B = B + A
Find the values of x, y, z if `[(x), (y – z), (z + 3)] + [(y), (4), (3)] = [(4), (8), (16)]`
Find the non-zero values of x satisfying the matrix equation
`x[(2x, 2),(3, x)] + 2[(8, 5x),(4, 4x)] = 2[(x^2 + 8, 24),(10, 6x)]`
Let A = `[(1, 2),(1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, 2)]` Show that (A – B)T = AT – BT
If A = `[("a", "b"),("c", "d")]` and I = `[(1, 0),(0, 1)]` show that A2 – (a + d)A = (bc – ad)I2
If A = `[(5, 2, 9),(1, 2, 8)]`, B = `[(1, 7),(1, 2),(5, -1)]` verify that (AB)T = BT AT
For the given matrix A = `[(1, 3, 5, 7),(2, 4, 6, 8),(9, 11, 13, 15)]` the order of the matrix AT is
Which of the following can be calculated from the given matrices A = `[(1, 2),(3, 4),(5, 6)]`, B = `[(1, 2, 3),(4, 5, 6),(7, 8, 9)]`,
(i) A2
(ii) B2
(iii) AB
(iv) BA
A = `[(3, 0),(4, 5)]`, B = `[(6, 3),(8, 5)]`, C = `[(3, 6),(1, 1)]` find the matrix D, such that CD – AB = 0
Find the values of p, q, r, and s if
`[("p"^2 - 1, 0, - 31 - "q"^3),(7, "r" + 1, 9),(- 2, 8, "s" - 1)] = [(1, 0, -4),(7, 3/2, 9),(-2, 8, -pi)]`
Consider the matrix Aα = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Show that `"A"_alpha "A"_beta = "A"_((alpha + beta))`
If A = `[(1, 0, 0),(0, 1, 0),("a", "b", - 1)]`, show that A2 is a unit matrix
Show that f(x) f(y) = f(x + y), where f(x) = `[(cosx, -sinx, 0),(sinx, cosx, 0),(0, 0, 1)]`
Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
`[(4, -2),(3, -5)]`
Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix
A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins and almonds. Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds. Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds. Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds. The cost of 50 gm of cashew nuts is ₹ 50, 50 gm of raisins is ₹ 10, and 50 gm of almonds is ₹ 60. What is the cost of each gift pack?
Let M = `[(0, -α),(α, 0)]`, where α is a non-zero real number an N = `sum_(k = 1)^49`M2k. If (I – M2)N = –2I, then the positive integral value of α is ______.
Let P = `[(3, -1, -2),(2, 0, alpha),(3, -5, 0)]`, where α ∈ R. Suppose Q = [qij] is a matrix satisfying PQ = kI3 for some non-zero k ∈ R. If q23 = `-k/8` and |Q| = `k^2/2`, then α2 + k2 is equal to ______.
