### 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)

and

(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

- Guess the inverse (I generally try this -- it is an immense waste of
time!)
- Write the inverse using a formula (not too hard)
- 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.