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

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The discussion focuses on mapping matrices to complex numbers in linear algebra, specifically addressing a matrix \( A \) with the property \( A^2 = -I \). This property suggests a correspondence to the complex number \( i \). The set \( K \) includes matrices of the form \( \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \), where \( a \) and \( b \) are real numbers. To establish that \( K \) is isomorphic to \( \mathbb{C} \), one must demonstrate that the mapping from \( \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \) to \( a + ib \) is a bijective isomorphism that preserves addition and multiplication. This involves mapping specific matrices to the corresponding complex numbers 1 and \( i \).
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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.
 

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