LA2.2.8

From Exampleproblems

Consider $A_{3\times 4}=I_{3\times 3}A_{3\times 4}I_{4\times 4}\,$

$\begin{bmatrix} 1 & 3 & 6 &-1 \\1 & 4 & 5 & 1 \\ 1 & 5 & 4 & 3\end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}AI_4\,$

Applying $R_2-R_1,R_3-R_1\,$ ,we get

$\begin{bmatrix} 1 & 3 & 6 & -1 \\ 0 & 1 & -1 & 2 \\ 0 & 2 & -2 & 4\end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{bmatrix}AI_4\,$

Now $R_3-2R_2\,$ implies

$\begin{bmatrix} 1 & 3 & 6 & -1 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0\end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 1 & -2 & 1\end{bmatrix}AI_4\,$

Applying $C_2-3C_1,C_3-6C_1,C_4-C_1\,$ to the above matrix,we get

$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0\end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 1 & -2 & 1\end{bmatrix}\times A \begin{bmatrix} 1 & -3 & -6 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}\,$

Now $C_3-C_2,C_4-2C_2\,$ implies

$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 1 & -2 & 1\end{bmatrix}\times A \begin{bmatrix} 1 & -3 & -9 & 7\\ 0 & 1 & 1 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}\,$