If A is `3xx4` matrix and B is a matrix such that A'B and BA' are both defined. Then B is of the type

Options

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

Answer

To solve the given problem, we need to determine the order of matrix B based on the conditions provided regarding the matrices A and B.

### Step-by-Step Solution:

1. **Identify the order of matrix A**:
- Given that matrix A is a \(3 \times 4\) matrix, it has 3 rows and 4 columns.

2. **Determine the order of \(A'\)**:
- The transpose of matrix A, denoted as \(A'\), will have its rows and columns swapped. Therefore, the order of \(A'\) will be \(4 \times 3\).

3. **Conditions for matrix multiplication**:
- For the product \(A'B\) to be defined, the number of columns in \(A'\) must equal the number of rows in B. Since \(A'\) is \(4 \times 3\), this means B must have 4 rows. Thus, if we denote the order of B as \(R \times S\), we have \(R = 4\).

4. **Determine the order of \(BA'\)**:
- For the product \(BA'\) to be defined, the number of columns in B must equal the number of rows in \(A'\). Since \(A'\) is \(4 \times 3\), B must have 3 columns. Therefore, we have \(S = 3\).

5. **Conclusion about the order of B**:
- From the above steps, we have determined that the order of matrix B must be \(4 \times 3\).

### Final Answer:
Thus, matrix B is of the type \(4 \times 3\).

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