Chapter 3: Q8P (page 141)

(a) Prove that. ${(AB)}^{t} = B^{t} A^{t}$ Hint: See $(9.10)$ .
(b) Verify (9.11), that is, show that (9.10) applies to a product of any number of matrices. Hint: Use (9.10)and (9.8).

Short Answer

Expert verified

a) It is proven that ${(AB)}^{t} = B^{t} A^{t} .$ .

b) It is proven that ${(ABCD)}^{T} = {(A (BCD))}^{T}$ .

Step by step solution

Property used.

Use this property to solve the problem.

$A^{t} = {(\overset{⇀}{A})}^{T} ABC = A (BC) = (AB) C$

Prove the given function (AB)t =BtAt ..

Taking Left hand side:

${(A B)}_{i k =}^{t} = (A B) k i (∵ A^{t} = {(A)}^{T}) = \underset{}{\sum_{j} A_{k j} B_{j i}} = \sum_{j} B_{j k}^{t} A_{i j}^{t} = \sum_{j} B_{i j}^{t} A_{j k}^{t}$

Solve further

$= {(B^{t} A^{t})}_{ik} {(AB)}^{t} = B^{t} A^{t}$

Hence proved.

Prove the given function(ABCD)T =(ABCD)T.

Taking Left hand side:

${(ABCD)}^{T} = {(A (BCD))}^{T} \{∵ ABC = A (BC) = (AB) C\} = {(BCD)}^{T} A^{T} \{use {(AB)}^{T} = B^{T} A^{T}\} = {(B (CD))}^{T} A^{T} = {(CD)}^{T} B^{T} A^{T} Solve the equation further = D^{T} C^{T} B^{T} A^{T} {(ABCD)}^{T} = D^{T} C^{T} B^{T} A^{T}$

Hence proved.

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(a) Prove that. ${(AB)}^{t} = B^{t} A^{t}$ Hint: See $(9.10)$ .
(b) Verify (9.11), that is, show that (9.10) applies to a product of any number of matrices. Hint: Use (9.10)and (9.8).

Short Answer

Step by step solution

Property used.

Prove the given function (AB)t =BtAt ..

Prove the given function(ABCD)T =(ABCD)T.

One App. One Place for Learning.

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(a) Prove that.(AB)t=BtAt Hint: See(9.10).(b) Verify (9.11), that is, show that (9.10) applies to a product of any number of matrices. Hint: Use (9.10)and (9.8).

Short Answer

Step by step solution

Property used.

Prove the given function (AB)t =BtAt ..

Prove the given function(ABCD)T =(ABCD)T.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Space Physics

Radiation

Force

Oscillations

Atoms and Radioactivity

Classical Mechanics

Study anywhere. Anytime. Across all devices.

(a) Prove that. ${(AB)}^{t} = B^{t} A^{t}$ Hint: See $(9.10)$ .
(b) Verify (9.11), that is, show that (9.10) applies to a product of any number of matrices. Hint: Use (9.10)and (9.8).