What characterizes a symmetric matrix?

A symmetric matrix is defined by the property that it is equal to its transpose. This means that for a matrix ( A ) to be considered symmetric, the condition ( A = A^T ) must hold true, where ( A^T ) represents the transpose of matrix ( A ). In practical terms, this implies that the element located at position ( (i, j) ) in the matrix is the same as the element at position ( (j, i) ). Therefore, symmetric matrices exhibit a certain kind of reflective symmetry across their diagonal.

Understanding this property is fundamental because symmetric matrices are important in various areas of mathematics and applications; they often arise in optimization problems, physics, and statistics, particularly in the context of covariance matrices.

The other statements do not accurately define symmetric matrices. While a symmetric matrix may have elements that are equal, this condition is not a general requirement for all symmetric matrices—symmetric matrices can have distinct elements in non-diagonal positions. The statement regarding a symmetric matrix being equal to its inverse only applies to specific cases, such as orthogonal symmetric matrices, but is not a defining feature of symmetry itself. Lastly, the assertion that a symmetric matrix always has a determinant of zero is incorrect;