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

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Can someone please help me solve Q2 in the attachment
 

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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.
 
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.
 
The world of 2\times 2 complex matrices is very colorful. They form a Banach-algebra, they act on spinors, they contain the quaternions, SU(2), su(2), SL(2,\mathbb C), sl(2,\mathbb C). Furthermore, with the determinant as Euclidean or pseudo-Euclidean norm, isu(2) is a 3-dimensional Euclidean space, \mathbb RI\oplus isu(2) is a Minkowski space with signature (1,3), i\mathbb RI\oplus su(2) is a Minkowski space with signature (3,1), SU(2) is the double cover of SO(3), sl(2,\mathbb C) is the...
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