I believe dogs have qualia because I believe all mammals share a common ancestor and thus it stands to reason that we, being somewhat related to dogs, must have somewhat similar experiences.

Although not everyone feels that way and many believe only humans have qualia.

More to the point though... It's a hard stretch convincing me that matrix multiplications lead to qualia. At which point in the multiplication is sensation felt?

Whereas a dog and a human presumably operate on principles we still don't understand, so there is room in this unknown space for qualia to exist.