Question

If \( A(\text{adj} A) = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix} \), then the value of \( |A| + |\text{adj} A| \) is equal to :

• A. 5
• B. 25
• C. 125
• D. 30

Hint:

The determinant of the adjoint of a matrix is given by \( |\text{adj} A| = |A|^{n-1} \), where \( n \) is the order of the matrix.

Answer 1

The Correct Option is D

Step 1: General formula for adjoint of a matrix.
We know that for any square matrix \( A \) of order \( n \), the product of \( A \) and its adjoint \( \text{adj} A \) satisfies the relation:
\[ A (\text{adj} A) = |A| I_n \] where \( I_n \) is the identity matrix of order \( n \), and \( |A| \) is the determinant of the matrix \( A \).

Step 2: Apply the given matrix.
In this case, the matrix \( A(\text{adj} A) \) is given as: \[ A (\text{adj} A) = \begin{bmatrix} 5 & 0 & 0 0 & 5 & 0 0 & 0 & 5 \end{bmatrix} \]
This matrix is a scalar multiple of the identity matrix. So, comparing this with \( A (\text{adj} A) = |A| I_3 \), we can conclude that: \[ |A| = 5 \]

Step 3: Formula for \( |\text{adj} A| \).
The determinant of the adjoint of a matrix is related to the determinant of the matrix by the following formula:
\[ |\text{adj} A| = |A|^{n-1} \]
where \( n \) is the order of the matrix. In this case, the matrix \( A \) is a 3x3 matrix, so \( n = 3 \). Therefore:
\[ |\text{adj} A| = |A|^{3-1} = |A|^2 = 5^2 = 25 \]

Step 4: Calculate \( |A| + |\text{adj} A| \).
Now, we can calculate the sum:
\[ |A| + |\text{adj} A| = 5 + 25 = 30 \]

Step 5: Conclusion.
Therefore, the value of \( |A| + |\text{adj} A| \) is 30, and the correct answer is option (D).