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We know that a linear transformation T from ${\mathrm{鈩�}}^4$to${\mathrm{鈩�}}^4$ is given by

$T\left(\stackrel{鈫�}{x}\right)=A\stackrel{鈫�}{x}$

Where A is the standard matrix of a linear transformation.

Given that

$$\begin{gathered} T\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=x_1\left[\begin{array}{c}22\\ 鈭�16\\ 8\\ 5\end{array}\right]+x_2\left[\begin{array}{c}13\\ 鈭�3\\ 9\\ 4\end{array}\right]+x_3\left[\begin{array}{c}8\\ 鈭�2\\ 7\\ 3\end{array}\right]+x_4\left[\begin{array}{c}3\\ 鈭�2\\ 2\\ 1\end{array}\right] \\ 鈬�A=\left[\begin{array}{cccc}22& 13& 8& 3\\ 鈭�16& 鈭�3& 鈭�2& 鈭�2\\ 8& 9& 7& 2\\ 5& 4& 3& 1\end{array}\right] \end{gathered}$$

To check the matrix A is invertible or not makethem$\left[A\left|I_n\right.\right]$ and compute rref$\left[A\left|I_n\right.\right]$

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

We have

$A=\left[\begin{array}{cccc}22& 13& 8& 3\\ 鈭�16& 鈭�3& 鈭�2& 鈭�2\\ 8& 9& 7& 2\\ 5& 4& 3& 1\end{array}\right]$

Then

$鈬�\left[A\left|I_4\right.\right]=\left[\begin{array}{cccc}22& 13& 8& 3\\ 鈭�16& 鈭�3& 鈭�2& 鈭�2\\ 8& 9& 7& 2\\ 5& 4& 3& 1\end{array}\left|\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right.\right]$

Applying row operations, we get

$\left[\left.\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right|\begin{array}{cccc}1& 鈭�2& 9& 鈭�25\\ 鈭�2& 5& 鈭�22& 60\\ 4& 鈭�9& 41& 鈭�112\\ 鈭�9& 17& 80& 222\end{array}\right]=\left[I_n\left|B\right.\right]$

This shows that the matrix A is invertible and its inverse is

$A^{鈭�1}=\left[\begin{array}{cccc}1& 鈭�2& 9& 鈭�25\\ 鈭�2& 5& 鈭�22& 60\\ 4& 鈭�9& 41& 鈭�112\\ 鈭�9& 17& 80& 222\end{array}\right]$

We know thata linear transformation T from ${\mathrm{鈩�}}^4$to${\mathrm{鈩�}}^4$ is given by

$T\left(\stackrel{鈫�}{x}\right)=A\stackrel{鈫�}{x}$

Where A is the standard matrix of a linear transformation.

Then the inverse of T is given by

$$\begin{gathered} T\left(\stackrel{鈫�}{y}\right)=A^{鈭�1}\stackrel{鈫�}{y} \\ 鈬�T\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\end{array}\right]=\left[\begin{array}{cccc}1& 鈭�2& 9& 鈭�25\\ 鈭�2& 5& 鈭�22& 60\\ 4& 鈭�9& 41& 鈭�112\\ 鈭�9& 17& 80& 222\end{array}\right]\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\end{array}\right] \\ 鈬�T\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\end{array}\right]=y_1\left[\begin{array}{c}1\\ 鈭�2\\ 4\\ 鈭�9\end{array}\right]+y_2\left[\begin{array}{c}鈭�2\\ 5\\ 鈭�9\\ 17\end{array}\right]+y_3\left[\begin{array}{c}9\\ 鈭�22\\ 41\\ 80\end{array}\right]+y_4\left[\begin{array}{c}鈭�25\\ 60\\ 鈭�112\\ 222\end{array}\right]. \end{gathered}$$

The inverse of the linear transformation

$$\begin{gathered} \mathit{T}\mathbf{\left[}\begin{array}{c}{\mathbf{x}}_{\mathbf{1}}\\ {\mathbf{x}}_{\mathbf{2}}\\ {\mathbf{x}}_{\mathbf{3}}\\ {\mathbf{x}}_{\mathbf{4}}\end{array}\mathbf{\right]}\mathbf{=}{\mathit{x}}_{\mathbf{1}}\mathbf{\left[}\begin{array}{c}\mathbf{22}\\ \mathbf{鈭�}\mathbf{16}\\ \mathbf{8}\\ \mathbf{5}\end{array}\mathbf{\right]}\mathbf{+}{\mathit{x}}_{\mathbf{2}}\mathbf{\left[}\begin{array}{c}\mathbf{13}\\ \mathbf{鈭�}\mathbf{3}\\ \mathbf{9}\\ \mathbf{4}\end{array}\mathbf{\right]}\mathbf{+}{\mathit{x}}_{\mathbf{3}}\mathbf{\left[}\begin{array}{c}\mathbf{8}\\ \mathbf{鈭�}\mathbf{2}\\ \mathbf{7}\\ \mathbf{3}\end{array}\mathbf{\right]}\mathbf{+}{\mathit{x}}_{\mathbf{4}}\mathbf{\left[}\begin{array}{c}\mathbf{3}\\ \mathbf{鈭�}\mathbf{2}\\ \mathbf{2}\\ \mathbf{1}\end{array}\mathbf{\right]}\mathbf{}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\mathbf{}{\mathit{鈩�}}^{\mathbf{4}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}{\mathit{鈩�}}^{\mathbf{4}}\mathbf{}\mathit{i}\mathit{s} \\ \mathit{T}\mathbf{\left[}\begin{array}{c}{\mathbf{y}}_{\mathbf{1}}\\ {\mathbf{y}}_{\mathbf{2}}\\ {\mathbf{y}}_{\mathbf{3}}\\ {\mathbf{y}}_{\mathbf{4}}\end{array}\mathbf{\right]}\mathbf{=}{\mathit{y}}_{\mathbf{1}}\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{鈭�}\mathbf{2}\\ \mathbf{4}\\ \mathbf{鈭�}\mathbf{9}\end{array}\mathbf{\right]}\mathbf{+}{\mathit{y}}_{\mathbf{2}}\mathbf{\left[}\begin{array}{c}\mathbf{鈭�}\mathbf{2}\\ \mathbf{5}\\ \mathbf{鈭�}\mathbf{9}\\ \mathbf{17}\end{array}\mathbf{\right]}\mathbf{+}{\mathit{y}}_{\mathbf{3}}\mathbf{\left[}\begin{array}{c}\mathbf{9}\\ \mathbf{鈭�}\mathbf{22}\\ \mathbf{41}\\ \mathbf{80}\end{array}\mathbf{\right]}\mathbf{+}{\mathit{y}}_{\mathbf{4}}\mathbf{\left[}\begin{array}{c}\mathbf{鈭�}\mathbf{25}\\ \mathbf{60}\\ \mathbf{鈭�}\mathbf{112}\\ \mathbf{222}\end{array}\mathbf{\right]}\mathbf{.} \end{gathered}$$