If A `A=[3-2 4-2]`and `I=[1 0 0 1]`, find k so that `A^2=k A-2I`.

Explanation

To solve the problem, we need to find the value of \( k \) such that \( A^2 = kA - 2I \), given the matrices:

\[ A = \begin{pmatrix} 3 & -2 \\ 4 & -2 \end{pmatrix}, \quad I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### 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:

- 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: Write the equation \( A^2 = kA - 2I \)

We substitute \( A \) and \( I \) into the equation:

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

Substituting the matrices:

\[ \begin{pmatrix} 1 & -2 \\ 4 & -4 \end{pmatrix} = k \begin{pmatrix} 3 & -2 \\ 4 & -2 \end{pmatrix} - 2 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Calculating \( -2I \):

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

Now, we have:

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

### Step 3: Set up equations by equating corresponding elements

We can equate the corresponding elements of the matrices:

1. From the first row, first column:
\[ 1 = 3k - 2 \quad \text{(1)} \]

2. From the first row, second column:
\[ -2 = -2k \quad \text{(2)} \]

3. From the second row, first column:
\[ 4 = 4k \quad \text{(3)} \]

4. From the second row, second column:
\[ -4 = -2k - 2 \quad \text{(4)} \]

### Step 4: Solve the equations

**From equation (2):**
\[ -2 = -2k \implies k = 1 \]

**From equation (3):**
\[ 4 = 4k \implies k = 1 \]

**From equation (1):**
\[ 1 = 3k - 2 \implies 3k = 3 \implies k = 1 \]

**From equation (4):**
\[ -4 = -2k - 2 \implies -2 = -2k \implies k = 1 \]

### Conclusion

In all cases, we find that:

\[ k = 1 \]

### Final Answer

The value of \( k \) is \( 1 \).

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