If A is a matrix of order `3xx3` and `|A|=10` , then `|adj.A|` is

Options

A. `0`
B. `10`
C. `100`
D. `1000`

Answer

To find the determinant of the adjoint of a matrix \( A \) of order \( 3 \times 3 \) given that \( |A| = 10 \), we can use the property of determinants related to the adjoint of a matrix.

### Step-by-Step Solution:

1. **Understand the Property of the Adjoint**:
The determinant of the adjoint of a matrix \( A \) of order \( n \) is given by the formula:
\[ |adj(A)| = |A|^{n-1} \]
where \( n \) is the order of the matrix.

2. **Identify the Order of the Matrix**:
In this case, the matrix \( A \) is of order \( 3 \times 3 \), so \( n = 3 \).

3. **Substitute the Values**:
Now, substituting the values into the formula:
\[ |adj(A)| = |A|^{3-1} = |A|^2 \]

4. **Use the Given Determinant**:
We know that \( |A| = 10 \). Therefore:
\[ |adj(A)| = 10^2 \]

5. **Calculate the Result**:
\[ |adj(A)| = 100 \]

### Final Answer:
Thus, the determinant of the adjoint of matrix \( A \) is \( 100 \).

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