If `A=[{:(,4,2),(,1,1):}]`, find `(A-2I) (A-3I)`.

Answer

To solve the problem, we need to find \((A - 2I)(A - 3I)\) where the matrix \(A\) is given as:

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

### Step 1: Find the Identity Matrix \(I\)
Since \(A\) is a \(2 \times 2\) matrix, the identity matrix \(I\) will also be a \(2 \times 2\) matrix:

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

### Step 2: Calculate \(2I\) and \(3I\)
Now, we will calculate \(2I\) and \(3I\):

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

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

### Step 3: Calculate \(A - 2I\)
Now, we will subtract \(2I\) from \(A\):

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

### Step 4: Calculate \(A - 3I\)
Next, we will subtract \(3I\) from \(A\):

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

### Step 5: Multiply \((A - 2I)(A - 3I)\)
Now we will multiply the two resulting matrices:

\[ (A - 2I)(A - 3I) = \begin{pmatrix} 2 & 2 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 1 & -2 \end{pmatrix} \]

To calculate the product, we use the formula for matrix multiplication:

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix} \]

Calculating each element:

- First row, first column: \(2 \cdot 1 + 2 \cdot 1 = 2 + 2 = 4\)
- First row, second column: \(2 \cdot 2 + 2 \cdot (-2) = 4 - 4 = 0\)
- Second row, first column: \(1 \cdot 1 + (-1) \cdot 1 = 1 - 1 = 0\)
- Second row, second column: \(1 \cdot 2 + (-1) \cdot (-2) = 2 + 2 = 4\)

Thus, we have:

\[ (A - 2I)(A - 3I) = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} \]

### Step 6: Final Result
The final result can be expressed as:

\[ (A - 2I)(A - 3I) = 4I \]