Given `A=[{:(,p,0),(,0,2):}], B=[{:(,0,-q),(,1,0):}], C=[{:(,2,-2),(,2,2):}] and BA=C^2.` Find the values of p and q.

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) given the matrices \( A \), \( B \), and \( C \) and the equation \( BA = C^2 \).

### Step 1: Define the matrices
Given:
\[ A = \begin{pmatrix} p & 0 \\ 0 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & -q \\ 1 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 2 & -2 \\ 2 & 2 \end{pmatrix} \]

### Step 2: Calculate \( BA \)
To find \( BA \), we multiply matrix \( B \) by matrix \( A \):
\[ BA = B \cdot A = \begin{pmatrix} 0 & -q \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} p & 0 \\ 0 & 2 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 0 \cdot p + (-q) \cdot 0 = 0 \)
- First row, second column: \( 0 \cdot 0 + (-q) \cdot 2 = -2q \)
- Second row, first column: \( 1 \cdot p + 0 \cdot 0 = p \)
- Second row, second column: \( 1 \cdot 0 + 0 \cdot 2 = 0 \)

Thus, we have:
\[ BA = \begin{pmatrix} 0 & -2q \\ p & 0 \end{pmatrix} \]

### Step 3: Calculate \( C^2 \)
Next, we calculate \( C^2 \):
\[ C^2 = C \cdot C = \begin{pmatrix} 2 & -2 \\ 2 & 2 \end{pmatrix} \cdot \begin{pmatrix} 2 & -2 \\ 2 & 2 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 2 \cdot 2 + (-2) \cdot 2 = 4 - 4 = 0 \)
- First row, second column: \( 2 \cdot (-2) + (-2) \cdot 2 = -4 - 4 = -8 \)
- Second row, first column: \( 2 \cdot 2 + 2 \cdot 2 = 4 + 4 = 8 \)
- Second row, second column: \( 2 \cdot (-2) + 2 \cdot 2 = -4 + 4 = 0 \)

Thus, we have:
\[ C^2 = \begin{pmatrix} 0 & -8 \\ 8 & 0 \end{pmatrix} \]

### Step 4: Set \( BA \) equal to \( C^2 \)
Now we equate \( BA \) and \( C^2 \):
\[ \begin{pmatrix} 0 & -2q \\ p & 0 \end{pmatrix} = \begin{pmatrix} 0 & -8 \\ 8 & 0 \end{pmatrix} \]

From this, we can set up the following equations:
1. \( -2q = -8 \)
2. \( p = 8 \)

### Step 5: Solve for \( p \) and \( q \)
From the first equation:
\[ -2q = -8 \implies 2q = 8 \implies q = \frac{8}{2} = 4 \]

From the second equation:
\[ p = 8 \]

### Final Answer
The values are:
\[ p = 8, \quad q = 4 \]