If `A=[(a,0,0),(0,a,0),(0,0,a)],a!=0` then | adj A| is equal to

Options

A. `a^(3)`
B. `a^(9)`
C. `a^(6)`
D. `a^(27)`

Answer

To find the value of |adj A| for the given matrix \( A = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix} \) where \( a \neq 0 \), we can follow these steps:

### Step 1: Determine the order of the matrix
The matrix \( A \) is a \( 3 \times 3 \) matrix. Therefore, the order \( n \) of the matrix is 3.

**Hint:** The order of a matrix is defined by the number of rows (or columns) it has.

### Step 2: Calculate the determinant of matrix A
The determinant of a diagonal matrix is the product of its diagonal elements. Thus, we can calculate:

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

**Hint:** For a diagonal matrix, the determinant can be found by multiplying the diagonal elements.

### Step 3: Use the property of adjoint
The determinant of the adjoint of a matrix is given by the formula:

\[ |\text{adj} A| = |A|^{n-1} \]

where \( n \) is the order of the matrix. Here, \( n = 3 \).

**Hint:** Remember that the adjoint of a matrix is related to its determinant through this specific formula.

### Step 4: Substitute the values into the formula
Now we can substitute the values we found:

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

### Step 5: Simplify the expression
Now, simplify the expression:

\[ |\text{adj} A| = a^{3 \cdot 2} = a^6 \]

### Conclusion
Thus, the value of \( |\text{adj} A| \) is \( a^6 \).

**Final Answer:** \( |\text{adj} A| = a^6 \)

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