Question

If the order of the matrix \( A \) is \( 3 \times 2 \), then the order of the matrix \( (A')' \) is:

• A. \( 2 \times 3 \)
• B. \( 3 \times 2 \)
• C. \( 2 \times 2 \)
• D. \( 3 \times 3 \)

Hint:

Applying the transpose operation an even number of times returns the matrix to its original state and original order.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The transpose of a matrix \( A \), denoted by \( A' \), is obtained by interchanging its rows and columns.
Step 2: Detailed Explanation:
1. Matrix \( A \) has order \( 3 \times 2 \) (3 rows, 2 columns).
2. Its transpose \( A' \) will have the order \( 2 \times 3 \) (2 rows, 3 columns).
3. Taking the transpose again, \( (A')' \), will swap the rows and columns of \( A' \) back.
4. This results in an order of \( 3 \times 2 \).
Mathematically, the property \( (A')' = A \) holds for any matrix.
Step 3: Final Answer:
The order of \( (A')' \) is the same as the order of \( A \), which is \( 3 \times 2 \).