I've been able to calculate the determinant of a matrix forever. If I didn't learn before my first linear algebra course in 1985, then I learned then. When I took linear algebra in my master's program, there might have been spent 2 minutes on the subject because it's assumed a student at this level would know how to calculate a determinant.
I remember calculating eigenvalues and eigenvectors back in 1985, and I calculated them again 4 years ago when I took linear algebra. I can calculate eigenvalues and eigenvectors till the cows come home.
Until today I didn't understand what a determinant "was", or what eigenvalues and eigenvectors "were". I knew I didn't understand, I assumed there had to be some physical representation for them, I wanted to understand. But I never got satisfactory answers.
Yesterday a former student--who is majoring in math!--dropped by school to visit. He saw that under our new standards, we're covering matrix operations in pre-calculus in a lot more depth than we did under the standards in place when he was a student. We got to talking about what we cover, and I mentioned that I still didn't know what a determinant really "was".
He pointed me to a YouTube video he found. I watched it this morning. Within the first 7 minutes I had a good, big picture understanding of what a determinant is/does. 7 minutes.
Then I looked at the related videos in the column on the right. There, near the top, was one on eigenvalues and eigenvectors. Less than 10 minutes later I had the understanding that had up until today had eluded me.
I like learning :)