Find the inverse of each of the matrices, if it exists.
Question:

Find the inverse of each of the matrices, if it exists.

$\left[\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right]$

Solution:

Let $A=\left[\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right]$

We know that A = IA

Applying $R_{1} \rightarrow R_{1}-\frac{1}{2} R_{2}$, we have:

$\left[\begin{array}{ll}0 & 0 \\ 4 & 2\end{array}\right]=\left[\begin{array}{cc}1 & -\frac{1}{2} \\ 0 & 1\end{array}\right] A$

Now, in the above equation, we can see all the zeros in the first row of the matrix on the L.H.S.

Therefore, $A^{-1}$ does not exist.