1

Is there a transformation that exists which will rotate a matrix of this form $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ into $\begin{bmatrix}c & a\\d & b\end{bmatrix}$. I have looked into this question and I am aware that this is definitely not a transpose of one matrix to another one.

motiur
  • 207

2 Answers2

3

Let the required matrix be $M$

Then $$M\left(\begin{matrix}a&b\\c&d\end{matrix}\right)=\left(\begin{matrix}c&a\\d&b\end{matrix}\right)$$ $$\implies M=\left(\begin{matrix}c&a\\d&b\end{matrix}\right)\left(\begin{matrix}a&b\\c&d\end{matrix}\right)^{-1}$$ $$=\left(\begin{matrix}c&a\\d&b\end{matrix}\right)\frac{1}{ad-bc}\left(\begin{matrix}d&-b\\-c&a\end{matrix}\right)$$ $$=\frac{1}{ad-bc}\left(\begin{matrix}cd-ac&-bc+a^2\\d^2-bc&-bd+ab\end{matrix}\right)$$

David Quinn
  • 34,121
2

If you multiply the original matrix by $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, and then take the transpose, you will get the result you want.

Wizact
  • 1,846