If `A=[{:(,2,-1),(,-1, 3):}]" evaluate "A^2-3A+3I`, where I is a unit matrix of order 2.

Answer

To evaluate the expression \( A^2 - 3A + 3I \), where \( A = \begin{pmatrix} 2 & -1 \\ -1 & 3 \end{pmatrix} \) and \( I \) is the identity matrix of order 2, we will follow these steps:

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

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

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

Calculating the elements:

- First row, first column: \( 2 \cdot 2 + (-1) \cdot (-1) = 4 + 1 = 5 \)
- First row, second column: \( 2 \cdot (-1) + (-1) \cdot 3 = -2 - 3 = -5 \)
- Second row, first column: \( (-1) \cdot 2 + 3 \cdot (-1) = -2 - 3 = -5 \)
- Second row, second column: \( (-1) \cdot (-1) + 3 \cdot 3 = 1 + 9 = 10 \)

Thus,

\[ A^2 = \begin{pmatrix} 5 & -5 \\ -5 & 10 \end{pmatrix} \]

### Step 2: Calculate \( 3A \)

Next, we calculate \( 3A \):

\[ 3A = 3 \cdot \begin{pmatrix} 2 & -1 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 6 & -3 \\ -3 & 9 \end{pmatrix} \]

### Step 3: Calculate \( 3I \)

The identity matrix \( I \) of order 2 is:

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

Now, calculate \( 3I \):

\[ 3I = 3 \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \]

### Step 4: Substitute into the expression \( A^2 - 3A + 3I \)

Now we substitute \( A^2 \), \( 3A \), and \( 3I \) into the expression:

\[ A^2 - 3A + 3I = \begin{pmatrix} 5 & -5 \\ -5 & 10 \end{pmatrix} - \begin{pmatrix} 6 & -3 \\ -3 & 9 \end{pmatrix} + \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \]

Calculating each element:

- First row, first column: \( 5 - 6 + 3 = 2 \)
- First row, second column: \( -5 - (-3) + 0 = -5 + 3 = -2 \)
- Second row, first column: \( -5 - (-3) + 0 = -5 + 3 = -2 \)
- Second row, second column: \( 10 - 9 + 3 = 1 + 3 = 4 \)

Thus, we have:

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

### Final Answer

The final result is:

\[ \begin{pmatrix} 2 & -2 \\ -2 & 4 \end{pmatrix} \]