Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra

What people said

Recommends AceldamA · 2021-03 · Mathematics Stack Exchange · on “Need help simplifying a set of equations (and understanding how to solve it)”

thank you so much for your thorough explanation. i was just watching a video [this video] by 3blue1brown about cramer's law, which is new territory for me, and you laid it out beautifully. thankfully i understand matrix math, but again i can't thank you enough.

Recommends Ben Grossmann · 2022-02 · Mathematics Stack Exchange · on “Where does the method for finding the inverse of a 3x3 matrix come from?”

This video [this video] gives a nice geometric explanation of Cramer's rule, which in turn can be used to derive the cofactor formula for the inverse of a matrix.

Cites Nεo Pλατo · 2020-02 · Mathematics Stack Exchange · on “How can I solve this linear transformation?”

If it's a linear transformation, you would want to turn it into a matrix then interpret it from there. Like so. $\displaystyle \begin{pmatrix} 2 & 1\\ 4 & 15 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} =\begin{pmatrix} -4 \\ -1 \end{pmatrix}$ Then Cramer's rule can take over from there. But you can still go easy with this one. The function is well defined anyway so: $2x+y=-4$ $4x+15y=-1$ $4x+2y=-8$ $13y=7$ $y=\dfrac{7}{13}$ $x=\dfrac{-4-\dfrac{7}{13}}{2}$ $=\dfrac{-59}{26}$ I think that should be it. But just because you mentioned linear algebra you can check out what I was saying [this video] there