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akalin.com
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For example, in the equation above, the first step would be to subtract (3) times the first equation from the second equation to yield [ \begin{aligned} y_1 &= x_1 + 2 \cdot x_2 \ y_2 - 3 \cdot y_1 &= -2 \cdot x_2\text{.} \end{aligned} ] Then, add the second equation back to the first equation: [ \begin{aligned} y_2 - 2 \cdot y_1 &= x_1 \ y_2 - 3 \cdot y_1 &= -2 \cdot x_2\text{.} \end{aligned} ] Finally, divide the second equation by (-2): [ \begin{aligned} y_2 - 2 \cdot y_1 &= x_1 \ (3/2) \cdot y_1 - (1/2) \cdot y_2 &= x_2\text{.} \end{aligned} ] This is equivalent to the matrix equation [ \begin{pmatrix} -2 & 1 \ 3/2 & -1/2 \end{pmatrix} \cdot \begin{bmatrix} y_1 \ y_2 \end{bmatrix} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\text{,} ] so [ M^{-1} = \begin{pmatrix} -2 & 1 \ 3/2 & -1/2 \end{pmatrix}\text{.} ]

I’m guessing this is some sort of MathML-in-JavaScript?

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Yeap, it’s KaTeX: https://khan.github.io/KaTeX/