If A is a singular matrix then Adj is

Options

A. non singular
B. singular
C. symmetric
D. skew symmetric

Answer

To solve the problem, we need to determine the adjoint of a singular matrix \( A \).

### Step-by-Step Solution:

1. **Understanding Singular Matrix**:
A matrix \( A \) is called singular if its determinant is zero. Therefore, we have:
\[ \text{det}(A) = 0 \]

2. **Using the Property of Adjoint**:
The relationship between a matrix \( A \) and its adjoint \( \text{adj}(A) \) is given by the property:
\[ A \cdot \text{adj}(A) = \text{det}(A) \cdot I_n \]
where \( I_n \) is the identity matrix of order \( n \).

3. **Substituting the Determinant**:
Since \( A \) is singular, we substitute \( \text{det}(A) = 0 \) into the equation:
\[ A \cdot \text{adj}(A) = 0 \cdot I_n = 0 \]
This means that the product of \( A \) and \( \text{adj}(A) \) results in the zero matrix.

4. **Conclusion about the Adjoint**:
From the equation \( A \cdot \text{adj}(A) = 0 \), we can conclude that:
\[ \text{adj}(A) \text{ is also a singular matrix.} \]
This is because if the product of two matrices results in the zero matrix, at least one of the matrices must be singular. Since \( A \) is singular, \( \text{adj}(A) \) must also be singular.

### Final Answer:
Thus, if \( A \) is a singular matrix, then \( \text{adj}(A) \) is also a singular matrix.

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