The matrix `A=[{:(1,2),(2,2):}]` satisfies which one of the following polynomial equations ?

Options

A. `A^(2)+3A+2I=0`
B. `A^(2)+3A-2I=0`
C. `A^(2)-3A-2I=0`
D. `A^(2)-3A+2I=0`

Answer

To determine which polynomial equation the matrix \( A = \begin{pmatrix} 1 & 2 \\ 2 & 2 \end{pmatrix} \) satisfies, we will perform the following steps:

### Step 1: Calculate \( A^2 \)
We need to compute \( A \cdot A \) (i.e., \( A^2 \)).

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

Calculating the elements:
- First row, first column: \( 1 \cdot 1 + 2 \cdot 2 = 1 + 4 = 5 \)
- First row, second column: \( 1 \cdot 2 + 2 \cdot 2 = 2 + 4 = 6 \)
- Second row, first column: \( 2 \cdot 1 + 2 \cdot 2 = 2 + 4 = 6 \)
- Second row, second column: \( 2 \cdot 2 + 2 \cdot 2 = 4 + 4 = 8 \)

Thus,

\[ A^2 = \begin{pmatrix} 5 & 6 \\ 6 & 8 \end{pmatrix} \]

### Step 2: Calculate \( 3A \)
Next, we compute \( 3A \).

\[ 3A = 3 \cdot \begin{pmatrix} 1 & 2 \\ 2 & 2 \end{pmatrix} = \begin{pmatrix} 3 \cdot 1 & 3 \cdot 2 \\ 3 \cdot 2 & 3 \cdot 2 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 6 & 6 \end{pmatrix} \]

### Step 3: Calculate \( 2I \)
Now, we compute \( 2I \), where \( I \) is the identity matrix.

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

### Step 4: Form the polynomial equation
Now we can form the polynomial equation. We want to check if:

\[ A^2 - 3A - 2I = 0 \]

Substituting the values we calculated:

\[ \begin{pmatrix} 5 & 6 \\ 6 & 8 \end{pmatrix} - \begin{pmatrix} 3 & 6 \\ 6 & 6 \end{pmatrix} - \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \]

Calculating the left-hand side:

\[ = \begin{pmatrix} 5 - 3 - 2 & 6 - 6 - 0 \\ 6 - 6 - 0 & 8 - 6 - 2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

Since the result is the zero matrix, we conclude that the matrix \( A \) satisfies the polynomial equation:

\[ A^2 - 3A - 2I = 0 \]

### Final Answer
The matrix \( A \) satisfies the polynomial equation \( A^2 - 3A - 2I = 0 \).

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