Question

Do symmetric matrices commute?

Answer 1

Hint: In this question, we need to state whether the symmetric matrices commute or not, and also if the answer is yes then we have to explain this. For this, we will understand the concept of symmetric matrices first and also commuting matrices.

Complete step-by-step solution:
Let us first understand the meaning of symmetric matrices.
In algebra, a symmetric matrix is a square matrix that does not change whenever its transpose is evaluated. That is, a symmetric matrix is one whose transpose is the same as the matrix on its own.
Now, we will see the concept of commuting matrices.
If the product of two matrices doesn't really depend on the sequence of multiplication, they commute. In other words, commuting matrices satisfy the condition such as \[A\centerdot B=B\centerdot A\]
Here, \[A\] and \[B\] are two matrices.
Now, after learning this, we can say that the symmetric matrices do commute. If an orthogonal matrix can diagonalize a collection of symmetric matrices at the same time, they should commute. A further remarkable feature of symmetric matrices is that if A and B are symmetric matrices, then AB is symmetric if and only if A and B commute, i.e., if \[A\centerdot B=B\centerdot A\]

Therefore, the symmetric matrices do commute.

Note: Here, students generally make mistakes in understanding the concept of symmetric matrices and their commutative property. So they may get confused with the result of the multiplication of two matrices. They frequently believe that the product of matrices A and B differs from the product of matrices B and A, where A and B are two symmetric matrices.