If `A=[(a,0),(0,(1)/(b))]`, then `A^(-1) =`

Options

A. `[(-a,0),(0,-(1)/(b))]`
B. `[(-(1)/(a),0),(0,-(1)/(b))]`
C. `[((1)/(a),0),(0,b)]`
D. `[(-a,0),(0,b)]`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} a & 0 \\ 0 & \frac{1}{b} \end{pmatrix} \), we will follow the steps below:

### Step 1: Identify the Matrix
The matrix \( A \) is given as:
\[ A = \begin{pmatrix} a & 0 \\ 0 & \frac{1}{b} \end{pmatrix} \]

### Step 2: Find the Determinant of Matrix \( A \)
The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} p & q \\ r & s \end{pmatrix} \) is calculated as:
\[ \text{det}(A) = ps - qr \]
For our matrix \( A \):
\[ \text{det}(A) = a \cdot \frac{1}{b} - 0 \cdot 0 = \frac{a}{b} \]

### Step 3: Find the Adjoint of Matrix \( A \)
The adjoint of a \( 2 \times 2 \) matrix \( \begin{pmatrix} p & q \\ r & s \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} s & -q \\ -r & p \end{pmatrix} \]
For our matrix \( A \):
\[ \text{adj}(A) = \begin{pmatrix} \frac{1}{b} & 0 \\ 0 & a \end{pmatrix} \]

### Step 4: Calculate the Inverse of Matrix \( A \)
The inverse of matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{\frac{a}{b}} \begin{pmatrix} \frac{1}{b} & 0 \\ 0 & a \end{pmatrix} = \frac{b}{a} \begin{pmatrix} \frac{1}{b} & 0 \\ 0 & a \end{pmatrix} \]

### Step 5: Simplify the Inverse
Now, we simplify the expression:
\[ A^{-1} = \begin{pmatrix} \frac{b}{a} \cdot \frac{1}{b} & 0 \\ 0 & \frac{b}{a} \cdot a \end{pmatrix} = \begin{pmatrix} \frac{1}{a} & 0 \\ 0 & b \end{pmatrix} \]

### Final Result
Thus, the inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} \frac{1}{a} & 0 \\ 0 & b \end{pmatrix} \]

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