If A `=[{:(a,0,0),(0,a,0),(0,0,a):}]`, then det `(adjA)` equals

Options

A. `a^(27)`
B. `a^(9)`
C. `a^(6)`
D. `a^(2)`

Answer

To find the determinant of the adjoint of the matrix \( A \) given by

\[ A = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix}, \]

we will follow these steps:

### Step 1: Calculate the determinant of \( A \)

The determinant of a diagonal matrix is the product of its diagonal elements. Therefore, we have:

\[ \text{det}(A) = a \cdot a \cdot a = a^3. \]

### Step 2: Use the property of determinants

The property of determinants states that for any \( n \times n \) matrix \( A \):

\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1}, \]

where \( n \) is the order of the matrix. In this case, \( n = 3 \).

### Step 3: Substitute the values into the property

Now we substitute the value of \( \text{det}(A) \) that we calculated in Step 1:

\[ \text{det}(\text{adj}(A)) = (a^3)^{3-1} = (a^3)^{2} = a^{6}. \]

### Final Result

Thus, the determinant of the adjoint of \( A \) is:

\[ \text{det}(\text{adj}(A)) = a^6. \]

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