# What does an inverse matrix abstracts?

I am trying to understand inverse matrixes more in depth.

I took the simplest example: 2 points in a 2d space and put it into a matrix.

$$\begin{pmatrix}5&7\\-2&3\end{pmatrix}$$

Calculating the inverse, we would get another matrix and another 2 points.

Where could this inverse be used / and for what purposes?

Can someone provide me with a trivial example (preferably in 2d)?

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A $2\times 2$ matrix corresponds to a map of the plane to itself. Not surprisingly, the inverse matrix of that matrix corresponds to the inverse map.

For instance, the matrix $$\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \phantom-\cos \theta \\ \end{bmatrix}$$ corresponds to a rotation of angle $\theta$ around the origin. The inverse matrix is $$\begin{bmatrix} \phantom-\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{bmatrix}$$ and corresponds to a rotation of angle $-\theta$ around the origin.

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The correspondence being that the matrix $A$ corresponds to the map which takes $x$ in the plane to $Ax$. – Gerry Myerson May 7 '12 at 0:26

If you multiply a number by 3, then multiplying by $3^{-1}$ "undoes" the multiplication by 3.

Similarly, if you multiply by some matrix $M$, then multiplying by $M^{-1}$ "undoes" that multiplication.

Another way of saying this is that $3 \times 3^{-1} = 1$, and $M \times M^{-1} = I$, the identity matrix. Like the number 1, the matrix $I$ "does nothing".

Note that not every matrix has an inverse. It has to be a square matrix, for a start. You can think about it this way: If I multiply $x$ by $0$, then I get $0$, and there is nothing I can possibly multiply that which will give me $x$ back. Likewise, there are matricies which cannot be "undone".

More concretely, you can think of a matrix as a coordinate transformation. So if $M$ is a matrix representing a rotation 60 degrees clockwise, then $M^{-1}$ would be a rotation 60 degrees anti-clockwise. And so on.

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The answers already posted are quite nice, and do a good job of answering your general question of what a matrix inverse really represents. Let me look at your specific example in more depth, though, because the column-vector interpretation of matrices is sometimes useful.

Let's say you picked two points in the plane, $p = (5,-2)$ and $q = (7,3)$, and stuck them together as columns of a $2\times2$ matrix $A = \begin{bmatrix}5 & 7 \\ -2 & 3\end{bmatrix}$. What this matrix represents is the unique linear transformation that maps the unit vectors $\hat x = (1,0)$ and $\hat y = (0,1)$ to $p$ and $q$ respectively. (Try it out yourself: calculate $A\hat x$ and $A\hat y$ and see what happens.)

So what does $A^{-1} \approx \begin{bmatrix}0.103 & -0.241 \\ 0.069 & 0.172\end{bmatrix}$ represent? As others have said, it represents the transformation that undoes the transformation caused by $A$: it maps $p$ and $q$ back to $\hat x$ and $\hat y$. As it turns out, this is also the transformation that is undone by $A$. So another way of looking at it is that its columns represent the points, $r = (0.103, 0.069)$ and $s = (-0.241, 0.172)$, that $A$ maps to $\hat x$ and $\hat y$ respectively. (This is because, for example, $AA^{-1}\hat x = \hat x$, but $A^{-1}\hat x$ = $r$; plug that in and you get $Ar = \hat x$, or in other words, that $A$ maps $r$ to $\hat x$.)

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