The sum of `[(2,-3),(5,-7)]` and its multiplicative inverse is

Options

A. `[(-5,0),(0,-5)]`
B. `[(0,-2),(-2,0)]`
C. `[(0,0),(0,0)]`
D. `[(0,+2),(-2,0)]`

Explanation

To solve the problem of finding the sum of the matrix \(\begin{pmatrix} 2 & -3 \\ 5 & -7 \end{pmatrix}\) and its multiplicative inverse, we will follow these steps:

### Step 1: Define the Matrix
Let the matrix \( A = \begin{pmatrix} 2 & -3 \\ 5 & -7 \end{pmatrix} \).

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

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

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

### Step 4: Calculate the Inverse of \( A \)
The multiplicative inverse \( A^{-1} \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Since \(\text{det}(A) = 1\), we have:
\[ A^{-1} = 1 \cdot \begin{pmatrix} -7 & 3 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} -7 & 3 \\ -5 & 2 \end{pmatrix} \]

### Step 5: Calculate the Sum \( A + A^{-1} \)
Now we will find the sum of the matrix \( A \) and its inverse \( A^{-1} \):
\[ A + A^{-1} = \begin{pmatrix} 2 & -3 \\ 5 & -7 \end{pmatrix} + \begin{pmatrix} -7 & 3 \\ -5 & 2 \end{pmatrix} \]

Adding the corresponding elements:
\[ = \begin{pmatrix} 2 + (-7) & -3 + 3 \\ 5 + (-5) & -7 + 2 \end{pmatrix} = \begin{pmatrix} -5 & 0 \\ 0 & -5 \end{pmatrix} \]

### Final Answer
The sum of the matrix and its multiplicative inverse is:
\[ \begin{pmatrix} -5 & 0 \\ 0 & -5 \end{pmatrix} \]