91Ó°ÊÓ

Let the given matrix is$\left[\begin{array}{cccc}1& 1& 2& 3\\ 0& -1& 0& 0\\ 2& 2& 5& 4\\ 0& 3& 0& 1\end{array}\right]$

To check the matrixis invertible or not make the$\left[\left.A\right|I_n\right]$and compute rref $\left[\left.A\right|I_n\right]$.

• If rref$\left[\left.A\right|I_n\right]$ is of the formthenis invertible matrix. And inverse of A is B .
• If rref $\left[\left.A\right|I_n\right]$is of the other form (i.e. its left hand side has other then identify matrix) then is not invertible matrix.

We have given a matrix

$12A=\begin{array}{cccc}1& 1& 2& 3\\ 0& -1& 0& 0\\ -2& 2& 5& 4\\ 3& 3& 0& 1\end{array}\begin{array}{}\\ \\ \\ \end{array}$

$\left[\left.A\right|I_n\right]=\left[\overline{)\begin{array}{cccc}1& 1& 2& 3\\ 0& -1& 0& 0\\ 2& 2& 5& 4\\ 0& 3& 0& 1\end{array}}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

To find the rref $\left[\left.A\right|I_n\right]$we will do the following row operations.

$\left[\left.A\right|I_n\right]=\left[\overline{)\begin{array}{cccc}1& 1& 2& 3\\ 0& -1& 0& 0\\ 2& 2& 5& 4\\ 0& 3& 0& 1\end{array}}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$R_3→R_3-2R_1$

$~\left[\overline{)\begin{array}{cccc}1& 1& 2& 3\\ 0& -1& 0& 0\\ 0& 0& 1& -2\\ 0& 3& 0& 1\end{array}}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ -2& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$$\begin{gathered} R_2→-R_2 \\ {R}_1→R_1-R_2 \\ {R}_4→R_4-3R_2 \end{gathered}$$

$~\left[\overline{)\begin{array}{cccc}1& 1& 2& 3\\ 0& -1& 0& 0\\ 0& 0& 1& -2\\ 0& 0& 0& 1\end{array}}\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ -2& 0& 1& 0\\ 0& 3& 0& 1\end{array}\right]$

$$\begin{gathered} R_1→R_1-2R_3 \\ ~\left[\overline{)\begin{array}{cccc}1& 0& 0& 7\\ 0& 1& 0& 0\\ 0& 0& 1& -2\\ 0& 0& 0& 1\end{array}}\begin{array}{cccc}5& 1& -2& 0\\ 0& -1& 0& 0\\ -2& 0& 1& 0\\ 0& 3& 0& 1\end{array}\right] \end{gathered}$$

$$\begin{gathered} R_1→R_1-7R_4 \\ {R}_3→R_3+2R_4 \end{gathered}$$

$~\left[\overline{)\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}}\begin{array}{cccc}5& -20& -2& -7\\ 0& -1& 0& 0\\ -2& 6& 1& 2\\ 0& 3& 0& 1\end{array}\right]=\left[\left.I_n\right|B\right]$

Hence, the given matrix $\left[\begin{array}{cccc}1& 1& 2& 3\\ 0& -1& 0& 0\\ 2& 2& 5& 4\\ 0& 3& 0& 1\end{array}\right]$ is invertible. The inverse of the matrix is$\left[\begin{array}{cccc}5& -20& -2& -7\\ 0& -1& 0& 0\\ -2& 6& 1& 2\\ 0& 3& 0& 1\end{array}\right]$