How Do You Map Matrices to Complex Numbers in Linear Algebra?

[Eric Machisi]

Can someone please help me solve Q2 in the attachment

 

[Opalg]

The matrix $A$ has the interesting property that its square is minus the identity matrix: $A^2 = \begin{bmatrix}-1&0\\0&-1\end{bmatrix}$. That suggests that it should correspond to the complex number $i$.

In fact, $K$ consists of all matrices of the form $\begin{bmatrix}a&-b\\b&a\end{bmatrix}$ where $a$ and $b$ are real numbers. To prove that $K$ is isomorphic to $\Bbb C$ you should show that the map taking $\begin{bmatrix}a&-b\\b&a\end{bmatrix}$ to $a+ib$ is an isomorphism. That is, it is a bijective map that preserves addition and multiplication.

 

[Kansas Boy]

It is sufficient to map $\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ to 1 and map $\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$ to i.