If `A+I=[(3,-2),(4,1)]` then `(A+I)(A-I)` is equal to

Options

A. `[(-5,-4),(8,-9)]`
B. `[(-5,4),(-8,9)]`
C. `[(5,4),(8,9)]`
D. `[(-5,-4),(-8,-9)]`

Explanation

To solve the problem, we start with the equation given:

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

We need to find the value of \( (A + I)(A - I) \).

### Step 1: Express \( A - I \)
We can express \( A - I \) in terms of \( A + I \):
\[ A - I = (A + I) - 2I \]
Where \( I \) is the identity matrix:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
Thus,
\[ 2I = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \]
So,
\[ A - I = \begin{pmatrix} 3 & -2 \\ 4 & 1 \end{pmatrix} - \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 3 - 2 & -2 - 0 \\ 4 - 0 & 1 - 2 \end{pmatrix} = \begin{pmatrix} 1 & -2 \\ 4 & -1 \end{pmatrix} \]

### Step 2: Calculate \( (A + I)(A - I) \)
Now we can calculate \( (A + I)(A - I) \):
\[ (A + I)(A - I) = \begin{pmatrix} 3 & -2 \\ 4 & 1 \end{pmatrix} \begin{pmatrix} 1 & -2 \\ 4 & -1 \end{pmatrix} \]

### Step 3: Perform the matrix multiplication
To multiply these matrices, we use the formula for matrix multiplication:
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix} \]

Applying this to our matrices:
- First element: \( 3 \cdot 1 + (-2) \cdot 4 = 3 - 8 = -5 \)
- Second element: \( 3 \cdot (-2) + (-2) \cdot (-1) = -6 + 2 = -4 \)
- Third element: \( 4 \cdot 1 + 1 \cdot 4 = 4 + 4 = 8 \)
- Fourth element: \( 4 \cdot (-2) + 1 \cdot (-1) = -8 - 1 = -9 \)

Thus, we have:
\[ (A + I)(A - I) = \begin{pmatrix} -5 & -4 \\ 8 & -9 \end{pmatrix} \]

### Final Result
The final result is:
\[ (A + I)(A - I) = \begin{pmatrix} -5 & -4 \\ 8 & -9 \end{pmatrix} \]