# Background

Let $$A = \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$$ be a matrix in $$\mathbb{R}^{2 \times 2}$$. While matrices are often used to represent a variety of linear transformations, including rotations, here I am transforming the matrix itself.

A classic way to introduce group transformations is via the following 8 transformations of a square with distinct corners:

• Rotation of 0 degrees: $$R_0(A)=\left[\begin{matrix}a & b\\c & d\end{matrix}\right]$$
• Rotation of 90 degrees: $$R_{90}(A)=\left[\begin{matrix}b & d\\a & c\end{matrix}\right]$$
• Rotation of 180 degrees: $$R_{180}(A)=\left[\begin{matrix}d & c\\b & a\end{matrix}\right]$$
• Rotation of 270 degrees: $$R_{270}(A)=\left[\begin{matrix}c & a\\d & b\end{matrix}\right]$$
• Flipping about a horizontal axis: $$H(A)=\left[\begin{matrix}c & d\\a & b\end{matrix}\right]$$
• Flipping about a vertical axis: $$V(A)=\left[\begin{matrix}b & a\\d & c\end{matrix}\right]$$
• Flipping about the main diagonal: $$D(A)=\left[\begin{matrix}a & c\\b & d\end{matrix}\right]$$
• Flipping about the other diagonal: $$D^\prime(A)= \left[\begin{matrix}d & b\\c & a\end{matrix}\right]$$

Considering the compositions of all pairs of operations yields an 8 by 8 Cayley table. Thinking of my matrix $$A$$ as the square to be rotated, I created a similar table except the entries are the determinants of the resulting matrices after applying a composition of two of the above operations which I have represented with matrix $$B$$ below. The order of the rows and columns follow the order of the transformations as they are listed above.

$$B = \left[\begin{matrix}a d - b c & - a d + b c & a d - b c & - a d + b c & - a d + b c & - a d + b c & a d - b c & a d - b c\\- a d + b c & a d - b c & - a d + b c & a d - b c & a d - b c & a d - b c & - a d + b c & a d - b c\\a d - b c & - a d + b c & a d - b c & - a d + b c & - a d + b c & - a d + b c & a d - b c & a d - b c\\- a d + b c & a d - b c & - a d + b c & a d - b c & a d - b c & a d - b c & - a d + b c & a d - b c\\- a d + b c & a d - b c & - a d + b c & a d - b c & a d - b c & a d - b c & - a d + b c & a d - b c\\- a d + b c & a d - b c & - a d + b c & a d - b c & a d - b c & a d - b c & - a d + b c & a d - b c\\a d - b c & - a d + b c & a d - b c & - a d + b c & - a d + b c & - a d + b c & a d - b c & a d - b c\\a d - b c & - a d + b c & a d - b c & - a d + b c & - a d + b c & - a d + b c & a d - b c & a d - b c\end{matrix}\right]$$

Finally I computed the determinant of $$B$$ to be $$\det (B) = 0$$.

# Question

Just as I started with 2 by 2 matrices and got a final "determinant of the matrix of determinants of the transformations of the group applied to the original matrix", will I also such a determinant of zero for any n by n matrix?

# Edits

## 1

Starting with $$A = \left[\begin{matrix}x_{0} & x_{1} & x_{2}\\x_{3} & x_{4} & x_{5}\\x_{6} & x_{7} & x_{8}\end{matrix}\right]$$, I found that the matrix of determinants was

$$B = \left[\begin{matrix}x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6}\\- x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6}\\x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6}\\- x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6}\\- x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6}\\- x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6}\\x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6}\\x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & - x_{0} x_{4} x_{8} + x_{0} x_{5} x_{7} + x_{1} x_{3} x_{8} - x_{1} x_{5} x_{6} - x_{2} x_{3} x_{7} + x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6} & x_{0} x_{4} x_{8} - x_{0} x_{5} x_{7} - x_{1} x_{3} x_{8} + x_{1} x_{5} x_{6} + x_{2} x_{3} x_{7} - x_{2} x_{4} x_{6}\end{matrix}\right]$$

And I similarly found $$\det(B) = 0$$ as was found in the 2 by 2 case.

## 2

I found the resulting determinant be zero in the 4 by 4 and 5 by 5 cases. Note that the transformations above are rotating/reflecting the matrix entries themselves, rather than vectors in a vector space.

• Getting any rotations etc. is just the same as multiplying matrix with a proper unitary matrix. Since determinant of any unitary matrix is 1 or -1 and since $det(A*B) = det(A)*det(B)$ the resulting matrix has as its entries determinant of the original matrix multiplied by 1 or -1. So, the whole problem reduces to determining determinant (pfffu) of this 1, and -1 matrix. It is unclear to me how you list rows/columns of your matrix however ... Jan 18 at 2:51

$$\DeclareMathOperator\id{id}$$ Let $$A$$ be an arbitrary $$n \times n$$ matrix. For each element $$\sigma \in D_{8},$$ we let $$\sigma \cdot A$$ denote the action of $$\sigma$$ on $$A$$ (so, if $$\sigma$$ is the rotation of the square by $$90$$ degrees counterclockwise, $$\sigma \cdot A$$ is the matrix obtained by rotating $$A$$ $$90$$ degrees counterclockwise).

Now, let $$B$$ be the matrix whose columns are labelled left to right with the elements of $$D_{8}$$ in some order (lets say $$\sigma_{1}, \sigma_{2}, \ldots, \sigma_{8}$$), and whose rows are labelled top to bottom with the elements of $$D_{8}$$ in the same order. If $$\sigma, \sigma'$$ are two elements of $$D_{8},$$ the entry in the $$\sigma$$ row and $$\sigma'$$ column of $$B$$ is $$\det(\sigma' \cdot (\sigma \cdot A)).$$

Note that rotation by $$0$$ degrees (the identity) and reflection across the main diagonal are elements of $$D_{8}$$; we let $$\id$$ and $$\varphi$$ denote these two, respectively. Furthermore, note that $$\id \cdot A = A,$$ while $$\varphi \cdot A = A^{T},$$ the transpose of $$A$$. We see that $$\det(\id \cdot A) = \det(A)$$, and that $$\det(\varphi \cdot A) = \det(A^{T}) = \det(A).$$

What are the entries of the $$\id$$ column of $$B$$? Well, for each $$\sigma_{i} \in D_{8},$$ we have $$\det(\id \cdot (\sigma \cdot A)) = \det(\sigma_{i} \cdot A),$$ so the entries of the $$\id$$ column of $$B$$ are just the determinants $$\det(\sigma_{i} \cdot A),$$ for $$i \in [0, 8].$$

Similarly, for each $$\sigma_{i} \in D_{8},$$ we have $$\det(\varphi \cdot (\sigma \cdot A)) = \det((\sigma_{i} \cdot A)^{T}) = \det(\sigma_{i} \cdot A),$$ so the entries of the $$\varphi$$ column of $$B$$ are also just the determinants $$\det(\sigma_{i} \cdot A),$$ for $$i \in [0, 8].$$

So, we see that $$B$$ has two identical columns, hence $$\det(B) = 0.$$ Note that this is true regardless of the size of $$A$$.

• Have you seen my answer ? Jan 18 at 11:22

In fact, the operations described by @ckefa can be given an all-matricial form, providing a direct access to their determinant. I am going to present them in $$n=2$$ dimensions for the benefit of simplicity, but they are valid in any dimension $$n$$:

Let

$$J=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \ \ A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$

Please note that anti-unit matrix $$J$$ has determinant $$-1$$ (valid in any dimension $$n$$).

The different operations you have can be given an all-matricial form ; using your notations, we have in particular:

• direct (= anti-clockwise) rotation by 90° is $$A \to R_{90}(M):=\color{red}{JA^T}=\begin{pmatrix}b&d\\a&c\end{pmatrix}.$$

• clockwise rotation by 90° is $$A \to R_{90}^{-1}(A):=\color{red}{A^TJ}.$$

• symmetry wrt to a vertical axis (= placing the columns in reverse order) is $$A \to V(A):=\color{red}{AJ}.$$

• symmetry wrt to a horizontal axis (= placing the lines in reverse order) is $$A \to H(A):=\color{red}{JA}.$$

Now, your matrix of determinants is more easily computable...

• For the "mattress group" itself, see this nice article bit-player.org/wp-content/extras/bph-publications/… Jan 18 at 20:54
• Besides, for the Cayley table of the group, do you know that the factorisation of its determinant has led to an important theory called the theory of group characters ? See here Jan 18 at 21:05