Final answer:
A matrix A is symmetric if it's equal to its transpose, meaning it has mirror symmetry about its main diagonal.
Step-by-step explanation:
To define what it means for a matrix A to be symmetric, we consider its transpose. A matrix A is said to be symmetric if and only if A is equal to its transpose, denoted by
This means that if you flip the matrix over its diagonal, it remains unchanged. Mathematically, this is represented as A(i,j) = A(j,i) for all i and j, where i and j are the row and column indices respectively.
In terms of bilaterally symmetric objects, it generally means that an object can be divided into two symmetrical parts across a unique plane, sometimes referred to as a mirror plane. Applying this concept to a matrix, the diagonal running from the top-left corner to the bottom-right corner serves as this 'mirror plane', making the entries symmetrical about this diagonal.