Symmetric Matrices and the Product of Two Matrices

Introduction to Linear Algebra at the Ohio State University quiz problems and solutions

Problem 111

Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.

(a) The product $AB$ is symmetric if and only if $AB=BA$.

(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.

 
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Hint.

A matrix $A$ is called symmetric if $A=A^{\trans}$.

In this problem, we need the following property of transpose:
Let $A$ be an $m\times n$ and $B$ be an $n \times r$ matrix. Then we have
\[(AB)^{\trans}=B^{\trans}A^{\trans}.\] (When you distribute transpose over the product of two matrices, then you need to reverse the order of the matrix product.)

Solution.

(a) The product $AB$ is symmetric if and only if $AB=BA$.

Suppose $AB$ is symmetric. This means that we have
\[ AB=(AB)^{\trans}=B^{\trans}A^{\trans}=BA.\] The second equality is a general property of transpose. In the last equality we used the assumption that $A, B$ are symmetric, that is, $A=A^{\trans}, B=B^{\trans}$. Thus we have $AB=BA$.

On the other hand, if $AB=BA$, then we have
\[AB=BA=B^{\trans}A^{\trans}=(AB)^{\trans}.\] Thus, we have $AB=(AB)^{\trans}$, hence $AB$ is symmetric.

If the product $AB$ is a diagonal matrix, then $AB=BA$.

Note that a diagonal matrix is symmetric. Hence the result follows from part (a).

10 True or False Problems about Matrices

Test your understanding about matrices with 10 True or False questions given in the post “10 True or False Problems about Basic Matrix Operations“.

Or try True or False problems about nonsingular, invertible matrices, and linearly independent vectors at “10 True of False Problems about Nonsingular / Invertible Matrices“.


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