If `A=[(b,0,0),(0,b,0),(0,0,b)]` then the value of `|A||adjA|` is

Options

A. `b^(3)`
B. `b^(9)`
C. `b^(6)`
D. `b^(8)`

Explanation

To solve the problem, we need to find the value of \(|A| \cdot |adj A|\) for the matrix \(A = \begin{pmatrix} b & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & b \end{pmatrix}\).

### Step-by-step Solution:

1. **Calculate the Determinant of Matrix A**:
The determinant of a diagonal matrix is the product of its diagonal elements. Here, the diagonal elements are \(b, b, b\).
\[ |A| = b \cdot b \cdot b = b^3 \]

2. **Find the Adjoint of Matrix A**:
For any square matrix \(A\), the adjoint (or adjugate) of \(A\) can be calculated using the formula:
\[ |adj A| = |A|^{n-1} \]
where \(n\) is the order of the matrix. Since \(A\) is a \(3 \times 3\) matrix, \(n = 3\).
\[ |adj A| = |A|^{3-1} = |A|^2 = (b^3)^2 = b^6 \]

3. **Calculate the Product of Determinant of A and Determinant of Adjoint A**:
Now, we can find \(|A| \cdot |adj A|\):
\[ |A| \cdot |adj A| = b^3 \cdot b^6 = b^{3+6} = b^9 \]

### Final Answer:
The value of \(|A| \cdot |adj A|\) is \(b^9\).

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