Let A be an n × n matrix with rank r(0 < r < n). Then Ax = 0 has p independent solutions, where p is

This question relates to the properties of matrices and the solutions to homogeneous systems of linear equations, specifically the equation Ax = 0.

Understanding Matrix Rank and Solutions

We are given an n × n matrix A with a rank r, where r is strictly between 0 and n (0 < r < n). We need to find the number of independent solutions, denoted by p, for the equation Ax = 0.

Applying the Rank-Nullity Theorem

The number of independent solutions to the equation Ax = 0 is determined by the dimension of the null space of the matrix A, also known as the nullity. The Rank-Nullity Theorem provides a fundamental relationship between the rank of a matrix and the dimension of its null space.

For an n × n matrix A, the Rank-Nullity Theorem states:

\**rank**(A) + \**nullity**(A) = n

In this context:

- n represents the total number of columns in the matrix A (which is also the number of variables in the vector x).
- rank(A) is given as r.
- nullity(A) is the dimension of the null space, which corresponds to the number of independent solutions, p.

Calculating the Number of Independent Solutions (p)

Substituting the given values into the Rank-Nullity Theorem:

$$r+p=n$$

To find p, we rearrange the equation:

$$p=n-r$$

Analyzing the Options

Based on our calculation using the Rank-Nullity Theorem, the number of independent solutions p is n - r.

Let's compare this with the given options:

1. r
2. n
3. n - r
4. n + r

Our calculated value matches option 3.

Conclusion

Therefore, for an n × n matrix A with rank r, the homogeneous system Ax = 0 has n - r independent solutions.