Show that we can find a matrix $\mathbf{B}$ such that $\mathbf{A} \mathbf{B} = \mathbf{I}$.

Question

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

I'm not sure how to prove this and I have no idea where to start. Help much appreciated, thanks!

Answer 1

Since for every $2D$ vector $v$, we can find $w$ such that $$Aw= v$$then let $w_1$ and $w_2$ such that $$Aw_1 = \begin{bmatrix} 1 \\ 0\end{bmatrix}$$ and $$Aw_2 = \begin{bmatrix} 0 \\ 1\end{bmatrix}$$ Hence $$A \begin{bmatrix} w_1 & w_2 \end{bmatrix} = I$$ So your matrix $B = \begin{bmatrix} w_1 & w_2 \end{bmatrix}$

Answer 2

Hint: Take $$ v_1 = \begin{pmatrix}1\\0\end{pmatrix}, \quad v_2 = \begin{pmatrix}0\\1\end{pmatrix}.$$ Then you can find vectors $w_1, w_2$ so that $Aw_j = v_j, \; j = 1,2.$

Answer 3

I think you are mainly asking for a hint about how to start, rather than a full solution - you should help people to give you the kind of answer which will help you most by making that explicit in your question. in general you get very little from just seeing someone else's solution to a question like this - you need to train your brain into the right ways of thinking about such problems.

Here is a hint: can you see how choosing suitable vectors for $v$ could help?

For future use - when you are given a general property like this, and also have a specific goal in mind, how can you choose values in the general property which relate to your specific question?