If `A = [(3,0,0),(0,3,0),(0,0,3)]`, then |A||adj A| =...

Answer

To solve the problem, we need to find the value of |A| * |adj A|, where \( A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix} \).

### Step 1: Calculate the Determinant of Matrix A

The determinant of a diagonal matrix is the product of its diagonal elements. Therefore, for matrix \( A \):

\[ |A| = 3 \cdot 3 \cdot 3 = 27 \]

### Step 2: Calculate the Determinant of the Adjoint of Matrix A

The determinant of the adjoint of a matrix \( A \) 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, \( n = 3 \).

Thus,

\[ |adj A| = |A|^{3-1} = |A|^2 \]

From Step 1, we already calculated \( |A| = 27 \). Therefore:

\[ |adj A| = 27^2 = 729 \]

### Step 3: Calculate |A| * |adj A|

Now, we can find the product:

\[ |A| \cdot |adj A| = 27 \cdot 729 \]

Calculating this gives:

\[ 27 \cdot 729 = 19683 \]

### Final Answer

Thus, the value of \( |A| \cdot |adj A| \) is:

\[ \boxed{19683} \]