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which proves that is symmetric.
Products and
Matrix products of the form and arise in a variety of applications. If A is an matrix, then is an matrix,
has size .
so the products and are both square matrices—the matrix has size , and the matrix
Such products are always symmetric since
EXAMPLE 6 The Product of a Matrix and Its Transpose Is Symmetric
Let A be the matrix
Then
Observe that and are symmetric as expected.
Later in this text, we will obtain general conditions on A under which and are invertible. However, in the special case
where A is square, we have the following result.
THEOREM 1.7.4
If A is an invertible matrix, then and are also invertible.
Proof Since A is invertible, so is by Theorem 1.4.10. Thus and are invertible, since they are the products of
invertible matrices.
Exercise Set 1.7
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