The inverse of a matrix `A=[[a, b], [c, d]]` is

Options

A. `[[d, -b], [-c, a]]`
B. `[[b, -a], [d, -c]]`
C. `(1)/(|A|)[[1, 0], [0, 1]]`
D. `(1)/(ad-bc)[[d, -b], [-c, a]]`

Answer

To find the inverse of a matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant \( \text{det}(A) \) of the matrix \( A \) is calculated using the formula:
\[ \text{det}(A) = ad - bc \]

### Step 2: Check if the Determinant is Non-Zero
For the inverse to exist, the determinant must be non-zero. Thus, we need to ensure that:
\[ ad - bc \neq 0 \]

### Step 3: Find the Adjoint of Matrix A
The adjoint (or adjugate) of the matrix \( A \) is given by:
\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

### Step 4: Use the Formula for the Inverse
The inverse of the matrix \( A \) can be calculated using the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{ad - bc} \cdot \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

### Step 5: Write the Final Expression for the Inverse
Thus, the inverse of the matrix \( A \) is:
\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

### Summary of the Inverse
The final expression for the inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{bmatrix} \frac{d}{ad - bc} & \frac{-b}{ad - bc} \\ \frac{-c}{ad - bc} & \frac{a}{ad - bc} \end{bmatrix} \]

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