If `A=[{:(3,-2),(4,-2):}]`, find x such that `A^(2)=xA - 2I`.

Answer

To solve the problem, we need to find the value of \( x \) such that:

\[ A^2 = xA - 2I \]

where \( A = \begin{pmatrix} 3 & -2 \\ 4 & -2 \end{pmatrix} \) and \( I \) is the identity matrix \( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \).

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \cdot A = \begin{pmatrix} 3 & -2 \\ 4 & -2 \end{pmatrix} \cdot \begin{pmatrix} 3 & -2 \\ 4 & -2 \end{pmatrix} \]

Calculating the elements of \( A^2 \):

- First row, first column:
\[ 3 \cdot 3 + (-2) \cdot 4 = 9 - 8 = 1 \]

- First row, second column:
\[ 3 \cdot (-2) + (-2) \cdot (-2) = -6 + 4 = -2 \]

- Second row, first column:
\[ 4 \cdot 3 + (-2) \cdot 4 = 12 - 8 = 4 \]

- Second row, second column:
\[ 4 \cdot (-2) + (-2) \cdot (-2) = -8 + 4 = -4 \]

Thus, we have:

\[ A^2 = \begin{pmatrix} 1 & -2 \\ 4 & -4 \end{pmatrix} \]

### Step 2: Calculate \( xA - 2I \)

Next, we calculate \( xA - 2I \):

\[ xA = x \begin{pmatrix} 3 & -2 \\ 4 & -2 \end{pmatrix} = \begin{pmatrix} 3x & -2x \\ 4x & -2x \end{pmatrix} \]

Now, calculate \( 2I \):

\[ 2I = 2 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \]

Now, we can find \( xA - 2I \):

\[ xA - 2I = \begin{pmatrix} 3x & -2x \\ 4x & -2x \end{pmatrix} - \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 3x - 2 & -2x \\ 4x & -2x - 2 \end{pmatrix} \]

### Step 3: Set \( A^2 = xA - 2I \)

Now, we set \( A^2 \) equal to \( xA - 2I \):

\[ \begin{pmatrix} 1 & -2 \\ 4 & -4 \end{pmatrix} = \begin{pmatrix} 3x - 2 & -2x \\ 4x & -2x - 2 \end{pmatrix} \]

### Step 4: Create equations from the matrix equality

From the matrix equality, we can create the following equations:

1. \( 3x - 2 = 1 \)
2. \( -2x = -2 \)
3. \( 4x = 4 \)
4. \( -2x - 2 = -4 \)

### Step 5: Solve the equations

**From equation 1:**
\[ 3x - 2 = 1 \implies 3x = 3 \implies x = 1 \]

**From equation 2:**
\[ -2x = -2 \implies x = 1 \]

**From equation 3:**
\[ 4x = 4 \implies x = 1 \]

**From equation 4:**
\[ -2x - 2 = -4 \implies -2x = -2 \implies x = 1 \]

All equations confirm that \( x = 1 \).

### Final Answer

Thus, the value of \( x \) is:

\[ \boxed{1} \]