If A is a non-singlular square matrix of order n, then the rank of A is

Options

A. equal to n
B. less than n
C. greater than n
D. none of these

Answer

To solve the question, we need to determine the rank of a non-singular square matrix \( A \) of order \( n \).

### Step-by-step Solution:

1. **Understanding Non-Singular Matrix**:
- A non-singular matrix is one whose determinant is not equal to zero. This implies that the matrix is invertible and has full rank.

**Hint**: Recall that a matrix is non-singular if its determinant is non-zero.

2. **Matrix Order**:
- The order of the matrix \( A \) is \( n \times n \), which means it has \( n \) rows and \( n \) columns.

**Hint**: Remember that the order of a matrix is defined by the number of rows and columns it has.

3. **Definition of Rank**:
- The rank of a matrix is defined as the maximum number of linearly independent rows (or columns) in the matrix. It can also be interpreted as the number of non-zero rows in its row echelon form.

**Hint**: The rank can be thought of as the dimension of the vector space spanned by its rows or columns.

4. **Maximum Rank of a Square Matrix**:
- For a square matrix of order \( n \), the maximum possible rank is \( n \). This means that if all rows (or columns) are linearly independent, the rank will be \( n \).

**Hint**: The rank of an \( n \times n \) matrix cannot exceed \( n \).

5. **Non-Zero Rows in Non-Singular Matrix**:
- Since \( A \) is non-singular, it cannot have any zero rows. If it had a zero row, the determinant would be zero, contradicting the non-singularity condition.

**Hint**: A non-singular matrix must have all rows (and columns) as non-zero to ensure linear independence.

6. **Conclusion**:
- Since all \( n \) rows of matrix \( A \) are non-zero and linearly independent, the rank of matrix \( A \) is equal to \( n \).

**Hint**: The rank of a non-singular \( n \times n \) matrix is always equal to \( n \).

### Final Answer:
The rank of matrix \( A \) is \( n \).