Question

Let A and B be two matrices and \(\gamma\) be any scalar, then :

• A. \((A^T)^T = A^T\)
• B. \((AB)^T = A^T B\)
• C. \((r A)^T = r^T A\)
• D. \((A + B)^T = A^T + B^T\)

Hint:

Remember the transpose rules:
- Addition preserves order: \((A+B)^T = A^T + B^T\).
- Multiplication reverses order: \((AB)^T = B^T A^T\).
- Double transpose cancels out: \((A^T)^T = A\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests the fundamental algebraic properties of matrix transposes.
The transpose of a matrix \(A\), denoted as \(A^T\), is formed by swapping its rows with its columns.

Step 2: Detailed Explanation:
Let us review the properties of matrix transposes to evaluate each option:
- Property 1 (Double Transpose): Taking the transpose of a transposed matrix returns the original matrix:
\[ (A^T)^T = A \]
Option (A) claims \((A^T)^T = A^T\), which is incorrect because it should equal \(A\).
- Property 2 (Product Transpose): The transpose of the product of two matrices is the product of their transposes in reverse order:
\[ (AB)^T = B^T A^T \]
Option (B) claims \((AB)^T = A^T B\), which is incorrect.
- Property 3 (Scalar Multiplication Transpose): The transpose of a scalar times a matrix is the scalar times the transpose of the matrix:
\[ (rA)^T = r A^T \]
Option (C) claims \((rA)^T = r^T A\), which is incorrect (since a scalar \(r\) does not have a transpose, and the matrix \(A\) must be transposed).
- Property 4 (Addition Transpose): The transpose of the sum of two matrices is equal to the sum of their individual transposes:
\[ (A + B)^T = A^T + B^T \]
This is a standard linear property of matrix operations. Therefore, Option (D) is correct.

Step 3: Final Answer:
The correct option is (D).