Question

Let \(A\) and \(B\) be two matrices and \(r\) 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:

Transpose distributes directly over addition/subtraction:
\[ (A \pm B)^T = A^T \pm B^T \]
But it reverses the order when distributing over multiplication:
\[ (AB)^T = B^T A^T \]

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question asks us to identify the correct algebraic property of matrix transposes among the given options.
The transpose of a matrix is formed by swapping its rows and columns.
Key Formula or Approach:
Let us review the standard properties of the transpose operation on matrices:
1. Double Transpose: \((A^T)^T = A\)
2. Scalar Multiple: \((r A)^T = r A^T\) (since \(r\) is a scalar, \(r^T = r\))
3. Sum of Matrices: \((A + B)^T = A^T + B^T\)
4. Product of Matrices: \((AB)^T = B^T A^T\)

Step 2: Detailed Explanation:

• Let us evaluate each option to find the correct statement:
- Option (A): \((A^T)^T = A^T\). This is incorrect because the transpose of a transpose returns the original matrix \(A\), so \((A^T)^T = A\).

• Evaluate Option (B): \((AB)^T = A^T B\).
This is incorrect because the transpose of a product reverses the order of multiplication: \((AB)^T = B^T A^T\).

• Evaluate Option (C): \((r A)^T = r^T A\).
This is incorrect. Since \(r\) is a scalar, \(r^T = r\). Thus, \((r A)^T = r A^T\). The option incorrectly leaves \(A\) untransposed and transposes the scalar.

• Evaluate Option (D): \((A + B)^T = A^T + B^T\).
This is a correct, fundamental property of matrix transposes, stating that the transpose of a sum is equal to the sum of the transposes.

Step 3: Final Answer:
The correct statement is \((A + B)^T = A^T + B^T\).