91影视

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

To check the matrix A is 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 form$\left[\left.I_n\right|B\right]$ thenis invertible matrix. And inverse ofis.
• 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

$\left[\begin{array}{cccc}1& 0& 0& 0\\ 2& 1& 0& 0\\ 3& 2& 1& 0\\ 4& 3& 2& 1\end{array}\right]$

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

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

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

$$\begin{gathered} R_2鈫�R_2-2R_1 \\ {R}_3鈫�R_3-3R_1 \\ {R}_4鈫�R_4-4R_1 \end{gathered}$$

$~\left[\overline{)\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 2& 1& 0\\ 0& 3& 2& 1\end{array}}\begin{array}{cccc}1& 0& 0& 0\\ -2& 1& 0& 0\\ -3& 0& 1& 0\\ -4& 0& 0& 1\end{array}\right]$

$$\begin{gathered} R_3鈫�R_3-2R_2 \\ {R}_4鈫�R_4-3R_2 \end{gathered}$$

$$\begin{gathered} ~\left[\overline{)\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 2& 1\end{array}}\begin{array}{cccc}1& 0& 0& 0\\ -2& 1& 0& 0\\ 1& -2& 1& 0\\ 8& -3& 0& 1\end{array}\right] \\ {R}_4鈫�R_4-2R_3 \\ ~\left[\overline{)\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 2& 1\end{array}}\begin{array}{cccc}1& 0& 0& 0\\ -2& 1& 0& 0\\ 1& -2& 1& 0\\ 8& 1& -2& 1\end{array}\right]=\left[\left.I_n\right|B\right] \end{gathered}$$

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