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# Long solution matrix part 2

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Let A be a $$2 \times 2$$ matrix. Suppose that for every two-dimensional vector v, there exists a two-dimensional vector w such that $$\mathbf{A} \mathbf{w} = \mathbf{v}$$
Show that we can find a matrix B such that $$\mathbf{A} \mathbf{B} = \mathbf{I}$$.

Mar 13, 2019

$$\exists w_1 \ni A w_1 = \begin{pmatrix}1\\0\end{pmatrix}\\ \exists w_2 \ni A w_2 = \begin{pmatrix}0\\1\end{pmatrix}\\ B = \begin{pmatrix}w_1 &w_2\end{pmatrix}$$