Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$A^{-1}=\left[\begin{array}{rr} {1}&{-1}\\ {-1}&{2}\end{array}\right]$
In order to calculate the inverse of an $n$ by $n$ non-singular matrix $A$, we will proceed with the following steps: Step -1. Transform the given matrix $A$ into this form $\left[A|I_{n}\right]$ as follows: $\left[A|I_{n}\right]$ = $\left[\begin{array}{ll|ll} {2}&{1}&{1}&{0}\\ {1}&{1}&{0}&{1}\end{array}\right]$ Step-2. Transform the matrix $\left[A|I_{n}\right]$ into reduced row-echelon form by using the row operations: $r_{1}\leftrightarrow r_{2}$ and $R_{2}=r_{2}-2r_{1}$ $\left[\begin{array}{rr|rr} {1}&{1}&{0}&{1}\\ {2}&{1}&{1}&{0}\end{array}\right]$ Multiply the above matrix by $-1$: $\left[\begin{array}{ll|ll} {1}&{1}&{0}&{1}\\ {0}&{-1}&{1}&{-2}\end{array}\right]$ Now, use the row operations: $R_{1}=r_{1}-r_{2}$ $\rightarrow\left[\begin{array}{cc|cc}{1}&{0}&{1}&{-1}\\{0}&{1}&{-1}&{2}\end{array}\right]$ Step-3: The reduced row-echelon form of $\left[A|I_{n}\right]$ will be represented as the identity matrix $I_{n}$ on the left of the vertical line and the $n$ by $n$ matrix on the right of the vertical line is the inverse of $A$. Thus, we have: $A^{-1}=\left[\begin{array}{rr} {1}&{-1}\\ {-1}&{2}\end{array}\right]$ (Note that if the identity matrix does not occur on the left, then the matrix $A$ does not have an inverse).