A hint for problem 4 of the Entrance Exam

Most of a message to me
... one question concerning problem 6. When in 640:250 Intro to Linear Algebra, they told us that the only time (at least the only time I can remember) that a 2x2 matrix is NOT invertible is when the determinant is equal to 0. In the problem you gave us, that leaves an infinite number of points where the matrix [3 A; 2 B] is invertible. Can you help narrow this down please?

My answer
Thanks for writing to me. Why shouldn't there be infinitely many such matrices which aren't invertible?

Hey, look at

(3 3)
(2 2)


(3 0)
(2 0)

I bet both of those aren't invertible! Let's see, for a 2-by-2 matrix, I think you can probably

  1. Guess the inverse (I generally try this -- it is an immense waste of time!)
  2. Write the inverse using a formula (not too hard)
  3. Use row reduction to get the inverse.
In all of these approaches you'll see an obstacle to writing the inverse, and that obstacle will turn out to be that the deterninant is NOT 0, exactly as you wrote. Just solve the problem as it is written, and don't bring any preconceptions of how many non-invertible matrices there should be. It WILL WORK out neatly.

I hope this message is helpful.

Maintained by greenfie@math.rutgers.edu and last modified 9/5/2005.