Question
Convert $\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]$ into an identity matrix by suitable row transformations.

Answer

Let $A=\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]$

By $R_2-2 R_1$, we get,

$A \sim\left[\begin{array}{rr}1 & -1 \\ 0 & 5\end{array}\right]$

By $\left(\frac{1}{5}\right) R_2$, we get,

$A \sim\left[\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right]$

$A \sim\left[\begin{array}{rr}1 & -1 \\0 & 1\end{array}\right]$

By $R_1+R_2$, we get,

$A \sim\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right]$

This is an identity matrix.

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