The meaning of Inverse Matrix

Solution 1:

When I took intro to Linear Algebra I found it useful to restrict myself to understanding $2\times2$ matrices. Once I understood a concept for those it was much easier to grasp the same concepts for higher dimensional square matrices. Rectangular matrices are not interesting in intro.

Think of matrix $A$ as a transformation. That is, take some vector in $x\in\mathbb{R}^2$. When you do $Ax$, you are in essence transforming the vector $x$ into something else. Of course, if $A$ is the identity matrix you can think of $Ax$ as $A$ transforming $x$ into itself.

So, $A$ could be a rotation transformation, or perhaps a dilation transformation, or any sort of transformation of $x$. When we say the inverse of $A$ exists, whom we denote $A^{-1}$, in essence we are saying:

Apply the transformation $A$ to $x$ to get some new vector $y$, possibly $x$ itself as described above. But, if we have $y$ and we want $x$ we need to undo what $A$ did to $x$. This is what $A^{-1}$ does. It takes $y$ in $A^{-1}y$ and undoes what $A$ did to get you back $x$. Notice that $A^{-1}$ itself is a transformation.

If $A$ rotated $x$ by 90 degrees clockwise to get $y$, then $A^{-1}$ will rotate $y$ 90 degreescounter-clockwise to get back $x$.

When $A^{-1}$ does not exist we mean, loosely speaking, that $A$ does not behave in a way that is predictable/reversible.

This concept is the same for higher dimension matrices, only the transformations are happening in 3D for $x\in\mathbb{R}^3$ and so on.

I hope this help. This is not a rigorous answer, as it is not meant to be, but it helped my intuition back then.

Solution 2:

I will give a basic definition/understanding, one that I learned in introductory Linear Algebra. For a matrix $A$, the inverse matrix $A^{-1}$ is a matrix that when multiplied by $A$ yields the Identity matrix of the vector space. $AA^{-1}=I$. $A^{-1}$ can be multiplied to the left or right of A, and still yield $I$. It can help in a scenario like $AB = AC$ where $A$ is invertible. $A^{-1}AB = A^{-1}AC$ which yields $B = C$. When a matrix doesn't have an inverse, it means just that. However, there are equivalent statements you should learn when $A$ is/is not invertible. For example, if $A$ is invertible then it is row-equivalent to $I$.