If A is a `3xx3` matrix and B is a matrix such that `A^TB and BA^(T)` are both defined, then order of B is

Options

A. `3xx4`
B. `3xx3`
C. `4xx4`
D. `4xx3`

Answer

To determine the order of matrix \( B \) given that \( A \) is a \( 3 \times 3 \) matrix and both \( A^T B \) and \( B A^T \) are defined, we can follow these steps:

### Step 1: Understand the dimensions of \( A \)
Matrix \( A \) is given as a \( 3 \times 3 \) matrix. Therefore, its transpose \( A^T \) will also be a \( 3 \times 3 \) matrix.

### Step 2: Define the order of matrix \( B \)
Let the order of matrix \( B \) be \( x \times y \). This means \( B \) has \( x \) rows and \( y \) columns.

### Step 3: Analyze the multiplication \( A^T B \)
For the multiplication \( A^T B \) to be defined, the number of columns in \( A^T \) must equal the number of rows in \( B \). Since \( A^T \) is \( 3 \times 3 \), we have:
\[ \text{Number of columns in } A^T = 3 = \text{Number of rows in } B = x \]
Thus, we conclude:
\[ x = 3 \]

### Step 4: Analyze the multiplication \( B A^T \)
For the multiplication \( B A^T \) to be defined, the number of columns in \( B \) must equal the number of rows in \( A^T \). Since \( A^T \) is \( 3 \times 3 \), we have:
\[ \text{Number of columns in } B = y = \text{Number of rows in } A^T = 3 \]
Thus, we conclude:
\[ y = 3 \]

### Step 5: Determine the order of matrix \( B \)
From the previous steps, we have determined that:
\[ x = 3 \quad \text{and} \quad y = 3 \]
Therefore, the order of matrix \( B \) is:
\[ B \text{ is of order } 3 \times 3 \]

### Final Answer
The order of matrix \( B \) is \( 3 \times 3 \).