Rank of Matrix AA

The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. In this case, we will determine the rank of the matrix AA, where A is a non-zero column vector of size n x 1.

Understanding the Matrix AA

The matrix AA is obtained by multiplying the column vector A with its transpose. The resulting matrix will have dimensions n x n, where n is the size of the column vector A. Each element of the matrix AA is obtained by multiplying the corresponding elements of A.

Determining the Rank

To determine the rank of matrix AA, we need to determine the maximum number of linearly independent rows or columns in the matrix.

Observation 1: Since A is a non-zero column vector, it will have at least one non-zero element.

Observation 2: When we multiply a non-zero column vector by its transpose, we obtain a matrix with all its columns equal to the original column vector A.

Conclusion: The matrix AA will have only one linearly independent column, which is the original column vector A.

Explanation:

- Since the matrix AA has only one linearly independent column, its rank is 1. This is because any other column of the matrix can be obtained as a linear combination of the original column vector A. For example, if we multiply the column vector A by a scalar, we obtain a column vector which is a multiple of A.

- The rank of a matrix is the maximum number of linearly independent rows or columns. In this case, since there is only one linearly independent column, the rank of matrix AA is 1.

- Option 'A' is the correct answer, as it represents a rank of 0 for the matrix AA.