Note Set 18d: Solving $\mathbf{A}\vec{x}=\vec{b}$ with determinants and Cramer's Rule

Rather surprisingly, we find that determinants allow us to find an exact solution for $\mathbf{A}\vec{x}=\vec{b}$ that always works if $\vec{b}$ is in the Column Space of $\mathbf{A}$. Things become completely crazy when we determine a formula for the inverse of $\mathbf{A}$. So our work should be done? No: determinants are hard to compute and the main use of Cramer's rule is for theoretical work and small systems.
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