Inverse Transformations Using Vectors and Matrices

Suppose we have the following $2\times 2$ matrix: $$A=\left[ \begin{array}{ccc} a & b \\ c & d \end{array} \right]$$ We want to find the inverse matrix of $A$: that is, we would like to find the matrix $A^{-1}$ such that the following holds: $$AA^{-1}=A^{-1}A=I$$ Denote the inverse of $A$ as follows: $$A^{-1}=\left[ \begin{array}{ccc} p & q \\ r & s \end{array} \right]$$ Then we have the following: $$\left[ \begin{array}{ccc} a & b \\ c & d \end{array} \right] \left[ \begin{array}{ccc} p & q \\ r & s \end{array} \right] =\left[ \begin{array}{ccc} 1 & 0 \\ 0 & 1 \end{array} \right]$$ Then we have the following four linear equations: $$ap+br=1$$ $$aq+bs=0$$ $$cp+dr=0$$ $$cq+ds=1$$ This is a system of four simultaneous linear equations with four variables, meaning that the ordered quadruple $(p,q,r,s)$ has a unique solution. You should work on solving it yourself, but I will not unnecessarily dive into the gritty details. Here are the solutions: $$p=\frac{d}{ad-bc}$$ $$q=\frac{-b}{ad-bc}$$ $$r=\frac{-c}{ad-bc}$$ $$s=\frac{a}{ad-bc}$$ Thus, we may now write the inverse matrix of $A$: $$\left[ \begin{array}{ccc} a & b \\ c & d \end{array} \right]^{-1}= \frac{1}{ad-bc}\left[ \begin{array}{ccc} d & -b \\ -c & a \end{array} \right]$$ What significance does this matrix hold in the world of transformations? It's exactly as expected! If the matrix $A$ defines some linear transformation, then the matrix $A^{-1}$ defines the inverse transformation.

Notice that the denominator, $\frac{1}{ad-bc}$, is exactly the reciprocal of the determinant of the new matrix. This gives us a very important result: $$\det(A^{-1})=\frac{1}{\det(A)}$$ This makes perfect sense. We know that a linear transformation magnifies areas by a factor equal to the absolute value of the determinant, and that the sign of the determinant determines whether the orientation is preserved or flipped. Well, the inverse transformation will now magnify areas by the reciprocal of the determinant to restore the area to its initial value. Also, it will either preserve the orientation or flip it again to set it back to what it was earlier.

Notice that a matrix does not have an inverse if it has a determinant of zero. This plays a significant role in solving linear equations, as this is when the linear equation system does not have a unique solution. Also, in the context of a transformation, such a matrix will map all points in the plane to a single line, thus reducing all areas to zero.