If `A=[{:(2,-1,3),(4,5,-6):}] and B=[{:(1,2),(3,4),(5,-6):}]` then

Options

A. only AB is defined
B. only BA is defined
C. AB and BA are both defined
D. AB and BA both are not defined

Answer

To determine whether the matrices \( A \) and \( B \) can be multiplied in both orders (i.e., \( AB \) and \( BA \)), we need to analyze the dimensions of both matrices.

1. **Identify the dimensions of matrix \( A \)**:
- Matrix \( A \) is given as:
\[ A = \begin{pmatrix} 2 & -1 & 3 \\ 4 & 5 & -6 \end{pmatrix} \]
- This matrix has 2 rows and 3 columns. Therefore, the order of matrix \( A \) is \( 2 \times 3 \).

2. **Identify the dimensions of matrix \( B \)**:
- Matrix \( B \) is given as:
\[ B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & -6 \end{pmatrix} \]
- This matrix has 3 rows and 2 columns. Therefore, the order of matrix \( B \) is \( 3 \times 2 \).

3. **Check if \( AB \) can be multiplied**:
- For the multiplication of two matrices \( AB \) to be valid, the number of columns in the first matrix \( A \) must equal the number of rows in the second matrix \( B \).
- Here, matrix \( A \) has 3 columns and matrix \( B \) has 3 rows.
- Since \( 3 = 3 \), the multiplication \( AB \) can be performed.

4. **Check if \( BA \) can be multiplied**:
- For the multiplication \( BA \) to be valid, the number of columns in matrix \( B \) must equal the number of rows in matrix \( A \).
- Here, matrix \( B \) has 2 columns and matrix \( A \) has 2 rows.
- Since \( 2 = 2 \), the multiplication \( BA \) can also be performed.

5. **Conclusion**:
- Both products \( AB \) and \( BA \) can be multiplied.

### Summary:
- The order of matrix \( A \) is \( 2 \times 3 \).
- The order of matrix \( B \) is \( 3 \times 2 \).
- \( AB \) can be multiplied (resulting in a \( 2 \times 2 \) matrix).
- \( BA \) can also be multiplied (resulting in a \( 3 \times 3 \) matrix).