Find |A (adjoint A)| and |adjoint A|, if `A=[(a,0,0),(0,a,0),(0,0,a)]`

Answer

To solve the problem, we need to find the values of |A (adjoint A)| and |adjoint A| for the matrix \( A = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix} \).

### Step 1: Calculate the Determinant of A
The determinant of a diagonal matrix is the product of its diagonal elements. Therefore, we have:

\[ |A| = a \cdot a \cdot a = a^3 \]

**Hint:** For a diagonal matrix, the determinant is simply the product of the diagonal entries.

### Step 2: Calculate the Adjoint of A
The adjoint of a matrix is the transpose of the cofactor matrix. For a diagonal matrix, the cofactors of the diagonal elements are the products of the other diagonal elements. Thus, the adjoint of \( A \) can be calculated as follows:

\[ \text{adjoint } A = \begin{pmatrix} a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2 \end{pmatrix} \]

**Hint:** The cofactor of a diagonal element in a diagonal matrix is the product of the other diagonal elements.

### Step 3: Calculate the Determinant of the Adjoint of A
Now, we need to find the determinant of the adjoint matrix:

\[ |\text{adjoint } A| = a^2 \cdot a^2 \cdot a^2 = (a^2)^3 = a^6 \]

**Hint:** Again, for a diagonal matrix, the determinant is the product of the diagonal entries.

### Step 4: Calculate |A (adjoint A)|
Next, we calculate the product \( A \cdot \text{adjoint } A \):

\[ A \cdot \text{adjoint } A = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix} \cdot \begin{pmatrix} a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2 \end{pmatrix} = \begin{pmatrix} a^3 & 0 & 0 \\ 0 & a^3 & 0 \\ 0 & 0 & a^3 \end{pmatrix} \]

The determinant of this resulting matrix is:

\[ |A \cdot \text{adjoint } A| = a^3 \cdot a^3 \cdot a^3 = a^9 \]

**Hint:** The determinant of the product of two matrices is the product of their determinants.

### Final Answers
- \( |A \cdot \text{adjoint } A| = a^9 \)
- \( |\text{adjoint } A| = a^6 \)