Question

Infinitely nested 2x2 matrix...

Answer 1

\[\left[\begin{matrix}\left[\begin{matrix}\left[\begin{matrix} & \\ & \end{matrix}\right] & \left[\begin{matrix} & \\ & \end{matrix}\right] \\ \left[\begin{matrix} & \\ & \end{matrix}\right] & \left[\begin{matrix} & \\ & \end{matrix}\right]\end{matrix}\right] & \left[\begin{matrix}\left[\begin{matrix} & \\ & \end{matrix}\right] & \left[\begin{matrix} & \\ & \end{matrix}\right] \\ \left[\begin{matrix} & \\ & \end{matrix}\right] & \left[\begin{matrix} & \\ & \end{matrix}\right]\end{matrix}\right] \\ \left[\begin{matrix}\left[\begin{matrix} & \\ & \end{matrix}\right] & \left[\begin{matrix} & \\ & \end{matrix}\right] \\ \left[\begin{matrix} & \\ & \end{matrix}\right] & \left[\begin{matrix} & \\ & \end{matrix}\right]\end{matrix}\right] & \left[\begin{matrix}\left[\begin{matrix} & \\ & \end{matrix}\right] & \left[\begin{matrix} & \\ & \end{matrix}\right] \\ \left[\begin{matrix} & \\ & \end{matrix}\right] & \left[\begin{matrix} & \\ & \end{matrix}\right]\end{matrix}\right]\end{matrix}\right]\]

So it's equal to its own determinant...What's the matrix? lol ok maybe this is too simple, but I wonder if someone has any fun ideas to go with this.

Answer 2

I wonder what happens if we try to multiply this by any scalar and try to write the result at the deepest level.

Answer 3

Of course there's no deepest level, so is this matrix multiplied by a scalar equal to the matrix itself? :P

Answer 4

lol, that was a joke.

Good question by the way.

Answer 5

Well since the matrix seems to be the zero matrix (I think) I don't know if multiplying it by a scalar will even matter!

Answer 6

Oh yeah, right. But don't take my replies seriously. Hehe.

Answer 7

Another fun to think about, what about having a matrix as the base of a logarithm? Is there some "natural" base like e is to scalars? Like, maybe the rotation matrix? \[\log_{\left[\begin{matrix}? & ? \\ ? & ?\end{matrix}\right]}(A)=B\]lol

Answer 8

Don't take my questions seriously! =P

Answer 9

\[A=\left[\begin{matrix}A & B \\ B & A\end{matrix}\right]\]\[B=\left[\begin{matrix}B & A \\ A & B\end{matrix}\right]\] Just a random thought, is this possible to be satisfied? If so, what are the matrices?