# Set of non-invertible matrices is unbounded

• Tigers64
In summary, the set of non-invertible matrices is unbounded because for any norm used, there exists a non-invertible matrix with a norm greater than any given bound.

#### Tigers64

Question:
How do I prove the set of non-invertible matrices is unbounded?

Attempt:
Let A be an element of set of non-invertible matrices.
det(A)=0
det(A)=0 is just the line y=0 if you have det(A) as the y-axis and the set of non-invertible matrices on the x-axis. y=0 is unbounded, so the set of non invertible matrices is unbounded?

Tigers64 said:
the set of non-invertible matrices on the x-axis
For this to make any sense at all, "the set of non-invertible matrices on the x-axis" would have to be a subset of "the set of real numbers"...

How do I prove the set of non-invertible matrices is unbounded?
Let's start with an easier question: can you find a non-invertible matrix whose norm is bigger than 10?

(p.s. what norm are you using?)

I guess the problem is that I don't know which norm to use, so I used det as the norm. How do you define a norm for matrices other than the det function?

How do you define a norm for matrices other than the det function?
There are infinitely many different norms you can define for matrices, several of which are in common use. This is a question I cannot answer for you -- you will have to check your homework problem / textbook / class notes to find out what norm you're supposed to be using.

(Incidentally, det isn't a norm. And even if it was, then the set of all non-invertible matrices would be bounded with respect to it)

Hurkyl said:
There are infinitely many different norms you can define for matrices, several of which are in common use. This is a question I cannot answer for you -- you will have to check your homework problem / textbook / class notes to find out what norm you're supposed to be using.

(Incidentally, det isn't a norm. And even if it was, then the set of all non-invertible matrices would be bounded with respect to it)
Since every non-invertible matrix has determinant 0, it would be very bounded!