# Set of non-invertible matrices is unbounded

## Main Question or Discussion Point

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?

## Answers and Replies

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Hurkyl
Staff Emeritus
Science Advisor
Gold Member
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?

Hurkyl
Staff Emeritus
Science Advisor
Gold Member
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)

HallsofIvy
Science Advisor
Homework Helper
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!