Set of all matrices with determinant 0, non-zero

Question

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes the set of nonzero real numbers.

For $f^{-1}(0)$, I understand that I'm looking for the set of all $n \times n$ matrices with determinant 0, but how do I express this in set theoretic notation that works for matrices of any order? I mean, for a matrix of order 2, $f^{-1}(0) = \{(a_{ij})| a_{11} a_{22} - a_{12} a_{21} =0, a_{ij} \in \mathbb R \}$. But this can't be extended to higher orders. I'm having the same issue with $f^{-1}(\mathbb R^*)$, where I'm looking for the set of all matrices with nonzero determinant. But how do I express this in set theoretic notation?

Is there any neat way that captures all such matrices, for both $f^{-1}(\mathbb R^*)$ and $f^{-1}(0)$?

Answer 1

We usually denote the set of $n\times n$ real matrices with non-zero determinant by $GL(n, \mathbb{R})$ - this is called the general linear group, hence the $GL$. Doing this we'd have $f^{-1}(\mathbb{R}^*) = GL(n, \mathbb{R})$ and $f^{-1}(0) = M(n, \mathbb{R})\setminus GL(n, \mathbb{R})$.

Answer 2

You would have $f^{-1}(0)=\{A\in M(n,\Bbb R):\det(A)=0\}$ And $f^{-1}(\Bbb R^*)=\{A\in M(n,\Bbb R):\det(A)\ne 0\}$

Or you could have:

$f^{-1}(0)=\{A\in M(n,\Bbb R):A \text{ has at least one eigenvalue $\lambda=0$}\}$

$f^{-1}(0)=\{A\in M(n,\Bbb R):A \text{ has all eigenvalue $\lambda\ne 0$}\}$

Answer 3

Since $\det A=0$ if and only if $A$ is invertible, we could write $f^{-1}(\Bbb R^*)$ and $f^{-1}(0)$ as \begin{align*} f^{-1}(\Bbb R^*) &= \{A\in M_n(\Bbb R):\exists B\in M_n(\Bbb R),AB= I\} \\ f^{-1}(0) &=\{A\in M_n(\Bbb R):\not\exists B\in M_n(\Bbb R),AB=I\} \end{align*}

Answer 4

• $f^{-1}(0)$ is the set of singular (non invertible) matrices of $M_n(\Bbb R)$ i.e. the set of matrices with rank strictly less than $n$ and

• $f^{-1}(\Bbb R^*)$ is the set of invertible matrices of $M_n(\Bbb R)$ i.e. the set of matrices with full rank denoted by $GL_n(\Bbb R)$.