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

$A^{-1}=\left[\begin{array}{rr} {1}&{-1/a}\\ {-1}&{2/a}\end{array}\right]$
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: $\left[A|I_{n}\right]$ = $\left[\begin{array}{ll|ll} {2}&{1}&{1}&{0}\\ {a}&{a}&{0}&{1}\end{array}\right]$ Step-2. Transform the matrix $\left[A|I_{n}\right]$ into reduced row echelon form by using Row operation: $R_{2}=r_{2}-ar_{1}$ $\left[\begin{array}{rr|rr} {1}&{1/2}&{1/2}&{0}\\ {a}&{a}&{0}&{1}\end{array}\right]$ Now, multiply Row-2 by $\dfrac{2}{a}$ to obtain: $\left[\begin{array}{cc|cc}{1}&{1/2}&{1/2}&{0}\\{0}&{a/2}&{-a/2}&{1}\end{array}\right]$ Finally, use Row operation: $R_{1}=r_{1}-\dfrac{r_2}{2}$ to obtain: $\left[\begin{array}{cc|cc}{1}&{0}&{1}&{-1/a}\\{0}&{1}&{-1}&{2/a}\end{array}\right]$ 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: $A^{-1}=\left[\begin{array}{rr} {1}&{-1/a}\\ {-1}&{2/a}\end{array}\right]$ (Note that if the identity matrix does not obtain on the left, then the matrix $A$ does have no inverse).