If A is a square matrix of order 3, such that `A(adjA)=10I`, then `|adjA|` is equal to

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 square matrix of order 3.

### 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 property of determinants**:
We know that:
\[ A \cdot \text{adj} A = |\text{det}(A)| \cdot I \]
This means that the left-hand side can also be expressed in terms of the determinant of \(A\).

3. **Equating the two expressions**:
From the above properties, we can equate:
\[ |\text{det}(A)| \cdot I = 10I \]
This implies:
\[ |\text{det}(A)| = 10 \]

4. **Finding the determinant of the adjoint**:
We use the formula for the determinant of the adjoint of a matrix:
\[ |\text{adj} A| = |\text{det}(A)|^{n-1} \]
where \(n\) is the order of the matrix. Since \(A\) is of order 3, we have \(n = 3\).

5. **Substituting the values**:
Now substituting the known values:
\[ |\text{adj} A| = |\text{det}(A)|^{3-1} = |\text{det}(A)|^2 = 10^2 = 100 \]

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

### Summary:
The determinant of the adjoint of matrix \(A\) is \(100\).