Given matrix `A=[{:(,4sin30^@ cos 0^@),(,cos 0^@ 4 sin 30^@):}] and B=[{:(,4),(,5):}]`. If AX=B.
write the order of matrix X.

Answer

To find the order of matrix \( X \) in the equation \( AX = B \), we need to analyze the orders of the matrices involved.

### Step-by-Step Solution:

1. **Identify the matrices**:
- Matrix \( A \) is given as:
\[ A = \begin{pmatrix} 4 \sin 30^\circ & \cos 0^\circ \\ \cos 0^\circ & 4 \sin 30^\circ \end{pmatrix} \]
- Matrix \( B \) is given as:
\[ B = \begin{pmatrix} 4 \\ 5 \end{pmatrix} \]

2. **Determine the order of matrix \( A \)**:
- Matrix \( A \) has 2 rows and 2 columns.
- Therefore, the order of matrix \( A \) is \( 2 \times 2 \).

3. **Determine the order of matrix \( B \)**:
- Matrix \( B \) has 2 rows and 1 column.
- Therefore, the order of matrix \( B \) is \( 2 \times 1 \).

4. **Set up the equation \( AX = B \)**:
- Let the order of matrix \( X \) be \( m \times n \).
- The multiplication of matrices \( A \) (which is \( 2 \times 2 \)) and \( X \) (which is \( m \times n \)) must result in matrix \( B \) (which is \( 2 \times 1 \)).

5. **Apply the rule of matrix multiplication**:
- The number of columns in the first matrix (matrix \( A \)) must equal the number of rows in the second matrix (matrix \( X \)).
- Therefore, since \( A \) has 2 columns, \( X \) must have 2 rows (i.e., \( m = 2 \)).
- The resulting matrix \( B \) has 1 column, so \( X \) must have 1 column (i.e., \( n = 1 \)).

6. **Conclusion**:
- The order of matrix \( X \) is \( 2 \times 1 \).

### Final Answer:
The order of matrix \( X \) is \( 2 \times 1 \).

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