If `A` is a matrix of order 3, such that `A(adj A)=10I,` then `|adj A|=`

Options

A. 1
B. 10
C. 100
D. 101

Answer

To solve the problem, we need to find the value of \(|\text{adj } A|\) given that \(A \cdot (\text{adj } A) = 10I\), where \(A\) is a \(3 \times 3\) matrix.

### Step-by-step Solution:

1. **Understanding the Given Equation**:
We start with the equation:
\[ A \cdot (\text{adj } A) = 10I \]
Here, \(I\) is the identity matrix of order 3.

2. **Using the Determinant Property**:
We know from matrix theory that:
\[ A \cdot (\text{adj } A) = \det(A) \cdot I \]
Therefore, we can equate the two expressions:
\[ \det(A) \cdot I = 10I \]

3. **Finding the Determinant of A**:
From the equation above, we can deduce:
\[ \det(A) = 10 \]

4. **Using the Formula for the Determinant of the Adjoint**:
The determinant of the adjoint of a matrix \(A\) is given by the formula:
\[ |\text{adj } A| = (\det A)^{n-1} \]
where \(n\) is the order of the matrix. Since \(A\) is a \(3 \times 3\) matrix, \(n = 3\).

5. **Calculating the Determinant of the Adjoint**:
Substituting the value of \(\det A\) into the formula:
\[ |\text{adj } A| = (\det A)^{3-1} = (\det A)^{2} = (10)^{2} = 100 \]

6. **Final Answer**:
Therefore, the value of \(|\text{adj } A|\) is:
\[ |\text{adj } A| = 100 \]