If `A=[[2, -3], [5, -7]]`, then `A+A^(-1)=`

Options

A. `[[1, 0], [0, 1]]`
B. `[[-5, 0], [0, -5]]`
C. `[[5, 0], [0, 5]]`
D. `[[4, 0], [0, -5]]`

Answer

To solve the problem \( A + A^{-1} \) where \( A = \begin{bmatrix} 2 & -3 \\ 5 & -7 \end{bmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of A
The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 2 \)
- \( b = -3 \)
- \( c = 5 \)
- \( d = -7 \)

Calculating the determinant:
\[ \text{det}(A) = (2)(-7) - (-3)(5) = -14 + 15 = 1 \]

### Step 2: Calculate the Adjoint of A
The adjoint of a 2x2 matrix \( A \) is given by:
\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
For our matrix \( A \):
\[ \text{adj}(A) = \begin{bmatrix} -7 & 3 \\ -5 & 2 \end{bmatrix} \]

### Step 3: Calculate the Inverse of A
The inverse of a matrix \( A \) is given by:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Since \( \text{det}(A) = 1 \):
\[ A^{-1} = 1 \cdot \begin{bmatrix} -7 & 3 \\ -5 & 2 \end{bmatrix} = \begin{bmatrix} -7 & 3 \\ -5 & 2 \end{bmatrix} \]

### Step 4: Calculate \( A + A^{-1} \)
Now we add \( A \) and \( A^{-1} \):
\[ A + A^{-1} = \begin{bmatrix} 2 & -3 \\ 5 & -7 \end{bmatrix} + \begin{bmatrix} -7 & 3 \\ -5 & 2 \end{bmatrix} \]
Adding the respective elements:
\[ = \begin{bmatrix} 2 + (-7) & -3 + 3 \\ 5 + (-5) & -7 + 2 \end{bmatrix} = \begin{bmatrix} -5 & 0 \\ 0 & -5 \end{bmatrix} \]

### Final Answer
Thus, \( A + A^{-1} = \begin{bmatrix} -5 & 0 \\ 0 & -5 \end{bmatrix} \).

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