If adj A = `[{:(a,0),(-1,b):}]` and ab `ne` 0, then what is the value of `|A^(-1)|` ?

Options

A. 1
B. ab
C. `1// sqrt(ab)`
D. `1//ab`

Explanation

To solve the problem, we need to find the value of \(|A^{-1}|\) given that \(\text{adj} A = \begin{pmatrix} a & 0 \\ -1 & b \end{pmatrix}\) and \(ab \neq 0\).

### Step-by-Step Solution:

1. **Understanding the relationship between adjoint and determinant**:
The determinant of the adjoint of a matrix \(A\) is related to the determinant of \(A\) by the formula:
\[ |\text{adj} A| = |A|^{n-1} \]
where \(n\) is the order of the matrix \(A\). For a \(2 \times 2\) matrix, \(n = 2\), so:
\[ |\text{adj} A| = |A|^{2-1} = |A| \]

2. **Calculating the determinant of adjoint A**:
We need to compute \(|\text{adj} A|\):
\[ \text{adj} A = \begin{pmatrix} a & 0 \\ -1 & b \end{pmatrix} \]
The determinant of a \(2 \times 2\) matrix \(\begin{pmatrix} p & q \\ r & s \end{pmatrix}\) is calculated as \(ps - qr\). Thus,
\[ |\text{adj} A| = ab - 0 \cdot (-1) = ab \]

3. **Using the relationship to find \(|A|\)**:
From step 1, we have:
\[ |\text{adj} A| = |A| \]
Therefore,
\[ |A| = ab \]

4. **Finding the determinant of the inverse of A**:
The determinant of the inverse of a matrix is given by:
\[ |A^{-1}| = \frac{1}{|A|} \]
Substituting the value of \(|A|\) we found:
\[ |A^{-1}| = \frac{1}{ab} \]

### Final Answer:
\[ |A^{-1}| = \frac{1}{ab} \]