Chapter 3: Q21P (page 142)

Show that the transpose of a sum of matrices is equal to the sum of the transposes. Also show that ${(M^{n})}^{T} = {(M^{T})}^{n}$ . Hint: Use (9.11)and (9.8).

Short Answer

Expert verified

The sum of the transposes is equal to the transpose of the sum of the matrices.

Step by step solution

Given information.

The equation, ${(M^{n})}^{T} = {(M^{T})}^{n}$ needs to be proven.

Step 2: Transpose of a matrix.

A matrix's transpose is determined by converting its rows into columns or columns into rows.

Show that the transpose of the sum of the matrix is equal to the sum of their transposes.

Assume A and B are two matrices. Now,

$(A + B)_{i j}^{T} = (A + B)_{i j} (A + B)_{i j} = A_{i j} + B_{i j} (A + B)_{i j}^{T} = A_{i j}^{T} + B_{i j}^{T} A_{i j}^{T} + B_{i j}^{T} = {(A^{T} + B^{T})}_{i j}$

Also,

$(A + B)_{i j}^{T} = {(A^{T} + B^{T})}_{i j} (A + B)^{T} = A^{T} + B^{T}$

Hence, from the above equation, the sum of the transposes is equal to the transpose of the sum of the matrices. Now,

${(M^{n})}^{T} = (M . M \dots M)^{T} (for n times) = (M (M \dots M))^{T}$

Since $A (B C) = (A B) C = A B C$ ,

$(M \dots M)^{T} M^{T} (∵ {(A B C D)}^{T} = D^{T} C^{T} B^{T} A^{T}) = (M (M \dots . M))^{T} M^{T} = ({(M \dots M)}^{T} M^{T}) M^{T}$

Solve further.

$= (M \dots M)^{T} M^{T} M^{T} = (M M)^{T} M^{T} \dots M^{T} = M^{T} M^{T} M^{T} \dots \dots M^{T} (n - times) = {(M^{T})}^{n}$

Hence,

${(M^{T})}^{n} = {(M^{n})}^{T}$

Hence, proved.

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Show that the transpose of a sum of matrices is equal to the sum of the transposes. Also show that ${(M^{n})}^{T} = {(M^{T})}^{n}$ . Hint: Use (9.11)and (9.8).

Short Answer

Step by step solution

Given information.

Step 2: Transpose of a matrix.

Show that the transpose of the sum of the matrix is equal to the sum of their transposes.

One App. One Place for Learning.

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Show that the transpose of a sum of matrices is equal to the sum of the transposes. Also show that(Mn)T=(MT)n. Hint: Use (9.11)and (9.8).

Short Answer

Step by step solution

Given information.

Step 2: Transpose of a matrix.

Show that the transpose of the sum of the matrix is equal to the sum of their transposes.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Famous Physicists

Fluids

Nuclear Physics

Electric Charge Field and Potential

Physics of Motion

Study anywhere. Anytime. Across all devices.

Show that the transpose of a sum of matrices is equal to the sum of the transposes. Also show that ${(M^{n})}^{T} = {(M^{T})}^{n}$ . Hint: Use (9.11)and (9.8).