Given `[{:(,2,1),(,-3,4):}]. X=[{:(,7),(,6):}]`. Write :
(i) the order of the matrix X.
(ii) the matrix X.

Answer

To solve the given problem step by step, we will address both parts: (i) finding the order of the matrix \( X \) and (ii) determining the matrix \( X \).

### Step 1: Determine the order of the matrix \( X \)

Given the equation:
\[ \begin{pmatrix} 2 & 1 \\ -3 & 4 \end{pmatrix} \cdot X = \begin{pmatrix} 7 \\ 6 \end{pmatrix} \]

1. The matrix on the left side, \( \begin{pmatrix} 2 & 1 \\ -3 & 4 \end{pmatrix} \), is a \( 2 \times 2 \) matrix (2 rows and 2 columns).
2. The matrix on the right side, \( \begin{pmatrix} 7 \\ 6 \end{pmatrix} \), is a \( 2 \times 1 \) matrix (2 rows and 1 column).

To multiply these matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Therefore, since the first matrix has 2 columns, the matrix \( X \) must have 2 rows.

Since the resulting matrix on the right side has 1 column, the matrix \( X \) must also have 1 column.

Thus, the order of the matrix \( X \) is:
\[ \text{Order of } X = 2 \times 1 \]

### Step 2: Determine the matrix \( X \)

Let us assume:
\[ X = \begin{pmatrix} A \\ B \end{pmatrix} \]

Now, substituting \( X \) into the equation:
\[ \begin{pmatrix} 2 & 1 \\ -3 & 4 \end{pmatrix} \cdot \begin{pmatrix} A \\ B \end{pmatrix} = \begin{pmatrix} 7 \\ 6 \end{pmatrix} \]

Performing the matrix multiplication:
\[ \begin{pmatrix} 2A + B \\ -3A + 4B \end{pmatrix} = \begin{pmatrix} 7 \\ 6 \end{pmatrix} \]

This gives us two equations:
1. \( 2A + B = 7 \) (Equation 1)
2. \( -3A + 4B = 6 \) (Equation 2)

Now, we will solve these equations simultaneously.

**From Equation 1:**
\[ B = 7 - 2A \]

**Substituting \( B \) into Equation 2:**
\[ -3A + 4(7 - 2A) = 6 \]
\[ -3A + 28 - 8A = 6 \]
\[ -11A + 28 = 6 \]
\[ -11A = 6 - 28 \]
\[ -11A = -22 \]
\[ A = 2 \]

**Substituting \( A \) back into Equation 1 to find \( B \):**
\[ B = 7 - 2(2) \]
\[ B = 7 - 4 \]
\[ B = 3 \]

Thus, the matrix \( X \) is:
\[ X = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \]

### Final Answers
(i) The order of the matrix \( X \) is \( 2 \times 1 \).
(ii) The matrix \( X \) is \( \begin{pmatrix} 2 \\ 3 \end{pmatrix} \).