Advertisements
Advertisements
Question
If A = `[(1, 0, 1),(3, 1, 2)]`, B = `[(2, 1, -4),(3, 5, -2)]` and C = `[(0, 2, 3),(-1, -1, 0)]`, verify that (A + 2B + 3C)T = AT + 2BT + 3CT
Solution
A + 2B + 3C
= `[(1, 0, 1),(3, 1, 2)] + 2[(2, 1, -4),(3, 5, -2)] + 3[(0, 2, 3),(-1, -1, 0)]`
= `[(1, 0, 1),(3, 1, 2)] + [(4, 2, -8),(6, 10, -4)] + [(0, 6, 9),(-3, -3, 0)]`
= `[(1 + 4 + 0, 0 + 2 + 6, 1 - 8 + 9),(3 + 6 - 3, 1 + 10 - 3, 2 - 4 + 0)]`
A + 2B + 3C = `[(5, 8, 2),(6, 8, -2)]`
∴ [A + 2B + 3C]T = `[(5, 6),(8, 8),(2, -2)]` ...(1)
Also, AT = `[(1, 3),(0, 1),(1, 2)]`, BT = `[(2, 3),(1, 5),(-4, -2)]`, CT = `[(0, -1),(2, -1),(3, 0)]`
∴ AT + 2BT + 3CT = `[(1, 3),(0, 1),(1, 2)] + 2[(2, 3),(1, 5),(-4, - 2)] + 3[(0, -1),(2, -1),(3, 0)]`
= `[(1, 3),(0, 1),(1, 2)] + [(4, 6),(2, 10),(-8, - 4)] + [(0, -3),(6, -3),(9, 0)]`
= `[(1 + 4 + 0, 3 + 6 - 3),(0 + 2 + 6, 1 + 10 - 3),(1 - 8 + 9, 2 - 4 + 0)]`
∴ AT + 2BT + 3CT = `[(5, 6),(8, 8),(2, -2)]` ...(2)
From (1) and (2),
[A + 2B + 3C]T = AT + 2BT + 3CT .
APPEARS IN
RELATED QUESTIONS
Find AT, if A = `[(1, 3),(-4, 5)]`
Find AT, if A = `[(2, -6, 1),(-4, 0, 5)]`
If A = `[(5, -3),(4, -3),(-2, 1)]`, Prove that (2A)T = 2AT
If A = `[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`, Prove that (3A)T = 3AT
If A = `[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]` where i = `sqrt(-1)` Prove that AT = – A
If A = `[(2, -3),(5, -4),(-6, 1)]`, B = `[(2, 1),(4, -1),(-3, 3)]` and C = `[(1, 2),(-1, 4),(-2, 3)]` then show that (A + B)T = AT + BT
If A = `[(2, -3),(5, -4),(-6, 1)]`, B = `[(2, 1),(4, -1),(-3, 3)]` and C = `[(1, 2),(-1, 4),(-2, 3)]` then show that (A – C)T = AT – CT
If A = `[(5, 4),(-2, 3)]` and B = `[(-1, 3),(4, -1)]`, then find CT , such that 3A – 2B + C = I, where I is the unit matrix of order 2
If A = `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find AT + 4BT
If A = `[(7, 3, 0),(0, 4, -2)]`, B = `[(0, -2, 3),(2, 1, -4)]` then find 5AT – 5BT
If A = `[(-1, 2, 1),(-3, 2, -3)]` and B = `[(2, 1),(-3, 2),(-1, 3)]`, prove that (A + BT)T = AT + B
Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where
A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`
Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where
A = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)]`
Express the following matrix as the sum of a symmetric and a skew symmetric matrix
`[(4, -2),(3, -5)]`
Express the following matrix as the sum of a symmetric and a skew symmetric matrix
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
If A = `[(2, -1),(3, -2),(4, 1)]` and B = `[(0, 3, -4),(2, -1, 1)]`, verify that (AB)T = BT AT
If A = `[(2, -1),(3, -2),(4, 1)]` and B = `[(0, 3, -4),(2, -1, 1)]`, verify that (BA)T = AT BT
If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, show that ATA = I, where I is the unit matrix of order 2
Select the correct option from the given alternatives:
Consider the matrices A = `[(4, 6, -1),(3, 0, 2),(1, -2, 5)]`, B = `[(2, 4),(0, 1),(-1, 2)]`, C = `[(3),(1),(2)]` out of the given matrix product ________
i) (AB)TC
ii) CTC(AB)T
iii) CTAB
iv) ATABBTC
Select the correct option from the given alternatives:
If A = `[(1, 2, 2),(2, 1, -2),("a", 2, "b")]` is a matrix satisfying the equation AAT = 9I, where I is the identity matrix of order 3, then the ordered pair (a, b) is equal to ________
Select the correct option from the given alternatives:
If A = `[(alpha, 2),(2, alpha)]` and |A3| = 125, then α = _______
Answer the following question:
If A = `[(cosalpha, -sinalpha),(sinalpha, cosalpha)]` and A + AT = I, where I is unit matrix 2 × 2, then find the value of α
Answer the following question:
If A = `[(2, 1, -3),(0, 2, 6)]`, B = `[(1, 0, -2),(3, -1, 4)]`, find ABT and ATB
Answer the following question:
If A = `[(3, -4),(1, -1)]`, prove that An = `[(1 + 2"n", -4"n"),("n", 1 - 2"n")]`, for all n ∈ N
Answer the following question:
If A = `[(costheta, sintheta),(-sintheta, costheta)]`, prove that An = `[(cos"n"theta, sin"n"theta),(-sin"n"theta, cos"n"theta)]`, for all n ∈ N
