0

I am studying matrices and I have the following doubt. Let's say we are in the $2$-dimensional world. A given matrix has $2$ columns, so it is $2 \times 2$. I interpret every column of the matrix as a point/vector, the same way 3blue1brown does it here. If I have:

\begin{bmatrix}2&0\\0&2\end{bmatrix}

its inverse is:

\begin{bmatrix}1/2&0\\0&1/2\end{bmatrix}

This, visually, is just the points $(2,0)$ and $(0,2)$ and its inverse are points $(\frac12, 0)$ and $(0, \frac12)$, if I draw the vectors as arrows like 3blue1brown, the inverse doesn't follow in an intuitive way, why is this?

The inverse doesn't make any sense to me. Visually the inverse is just a 'shorter' version of the original one.

Rodrigo de Azevedo
  • 23,223
Chicago1988
  • 169
  • 5
  • Possibly related – Rodrigo de Azevedo Apr 09 '25 at 16:52
  • 2
    These are scaling matrices. It's the same as multiplying the vector by $2$ or $\frac12$. Can you see why $2$ and $\frac12$ are inverse under multiplication? – CyclotomicField Apr 09 '25 at 16:55
  • 1
    This is an old post but your view of matrices is too narrow. They represent functions so a matrix inverse just represents the inverse of the function that the original matrix represented. Visually, if a vector $\vec x$ gets mapped to $\vec y$, then the inverse maps $\vec y$ to $\vec x$. – John Douma Apr 09 '25 at 17:19

0 Answers0