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Understanding the independent elements within matrices is vital before delving deeper. An independent element in a matrix is one that can be altered without affecting the matrix's inherent properties. In the case of symmetric matrices, these elements are particularly important.

For a symmetric matrix, which is defined by the property that it is equal to its transpose (\( A = A^T \)), the independent elements are found on the diagonal and the lower triangular portion of the matrix. This is because the values in the upper triangle are determined by those in the lower triangle. The unique characteristic of a symmetric matrix is that the elements across the diagonal mirror each other.

When calculating the number of independent elements in a symmetric matrix, we count each element on the main diagonal (which has 6 elements for a 6x6 matrix) once and then add the number of elements in the lower triangular part excluding the diagonal. For a 6x6 matrix, there are \( \frac{6 \times 5}{2} = 15 \) elements below the diagonal, resulting in a total of \( 6 + 15 = 21 \) independent elements.

Matrix symmetry is an innate property that makes certain types of matrices very predictable and defined. A matrix is considered symmetric if it is equal to its own transpose. This transpose is simply the matrix with its rows and columns interchanged.

The elegance of symmetric matrices is such that if you know the elements below the diagonal, you can effortlessly map out the entire matrix. This is because the corresponding elements above the diagonal will be the same. In mathematical notation, for a symmetric matrix \( A \), element \( a_{ij} \) will always equal \( a_{ji} \).

This intrinsic symmetry reduces the complexity when working with these matrices, as it limits the number of independent elements, drastically simplifying operations such as finding eigenvalues, computing determinants, and solving linear systems where symmetric matrices are involved.

Combinatorics is a fascinating area of mathematics dedicated to counting, arrangement, and combination of objects following specific rules. In the context of symmetric matrices, combinatorics plays a pivotal role.

Since each independent element in a symmetric matrix can either be a 0 or a 1, the task of finding the number of possible matrices is a combinatoric problem. Here, we use the simple principle of multiplication: if one event can occur in 'm' ways and another can occur independently in 'n' ways, then the two events can occur in \( m \times n \) ways.

Applying this principle to our matrix, each of the 21 independent elements has 2 options. Therefore, we can calculate the total number of distinct symmetric matrices by raising 2 to the power of the number of independent elements, resulting in \( 2^{21} \) possible matrices. This approach showcases how combinatorics underpins the principles for calculating permutations and combinations in matrix theory and beyond.