91Ó°ÊÓ

Consider the matrix $B\mathit{=}\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}$. such that $\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}A=A\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}$where

role="math" localid="1659847661790" $\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]$

Substitute the value $\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]$for A in the equation $\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}A=A\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}$as follows.

role="math" localid="1659847852946" $$\begin{gathered} \mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}A=A\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]} \\ \mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left[\begin{array}{ccc}\mathrm{2}{\mathrm{a}}_{11}& \mathrm{2}{\mathrm{a}}_{12}& \mathrm{2}{\mathrm{a}}_{13}\\ \mathrm{3}{\mathrm{a}}_{21}& \mathrm{3}{\mathrm{a}}_{22}& \mathrm{3}{\mathrm{a}}_{23}\\ \mathrm{3}{\mathrm{a}}_{31}& \mathrm{3}{\mathrm{a}}_{32}& \mathrm{3}{\mathrm{a}}_{33}\end{array}\right]\mathit{=}\mathit{\left[}\begin{array}{ccc}\mathit{2}{\mathit{a}}_{\mathit{11}}& \mathit{3}{\mathit{a}}_{\mathit{12}}& \mathit{3}{\mathit{a}}_{\mathit{13}}\\ \mathit{2}{\mathit{a}}_{\mathit{21}}& \mathit{3}{\mathit{a}}_{\mathit{22}}& \mathit{3}{\mathit{a}}_{\mathit{23}}\\ \mathit{2}{\mathit{a}}_{\mathit{31}}& \mathit{3}{\mathit{a}}_{\mathit{32}}& \mathit{3}{\mathit{a}}_{\mathit{33}}\end{array}\mathit{\right]} \end{gathered}$$

Compare the equation both side as follows.

$$\begin{gathered} 2{\mathrm{a}}_{11}=2{\mathrm{a}}_{11} \\ 2{\mathrm{a}}_{12}=3{\mathrm{a}}_{12} \\ 2{\mathrm{a}}_{13}=3{\mathrm{a}}_{13} \\ 3{\mathrm{a}}_{21}=2{\mathrm{a}}_{21} \end{gathered}$$

Further, compare the equation as follows.

$$\begin{gathered} 3{\mathrm{a}}_{22}=3{\mathrm{a}}_{22} \\ 3{\mathrm{a}}_{23}=3{\mathrm{a}}_{23} \\ 4{\mathrm{a}}_{31}=2{\mathrm{a}}_{31} \\ 4{\mathrm{a}}_{32}=3{\mathrm{a}}_{32} \end{gathered}$$

Further, compare the equation as follows.

$4{\mathrm{a}}_{33}=3{\mathrm{a}}_{33}$

Simplify the equationrole="math" localid="1659848047718" $2{\mathrm{a}}_{11}=2{\mathrm{a}}_{11}$as follows.

$$\begin{gathered} 2{\mathrm{a}}_{11}=2{\mathrm{a}}_{11} \\ {\mathrm{a}}_{11}={\mathrm{a}}_{11} \end{gathered}$$

Simplify the equation$2{\mathrm{a}}_{12}=3{\mathrm{a}}_{12}$as follows.

$$\begin{gathered} 2{\mathrm{a}}_{12}=4{\mathrm{a}}_{12} \\ {\mathrm{a}}_{12}=0 \end{gathered}$$

Simplify the equation$2{\mathrm{a}}_{13}=3{\mathrm{a}}_{13}$as follows.

$$\begin{gathered} 2{\mathrm{a}}_{13}=3{\mathrm{a}}_{13} \\ {\mathrm{a}}_{13}=0 \end{gathered}$$

Simplify the equation $3{\mathrm{a}}_{21}=2{\mathrm{a}}_{21}$as follows.

$$\begin{gathered} 3{\mathrm{a}}_{21}=2{\mathrm{a}}_{21} \\ {\mathrm{a}}_{21}=0 \end{gathered}$$

Simplify the equationrole="math" localid="1659848320356" $3{\mathrm{a}}_{22}=2{\mathrm{a}}_{22}$as follows.

$$\begin{gathered} 3{\mathrm{a}}_{22}=2{\mathrm{a}}_{22} \\ {\mathrm{a}}_{22}={\mathrm{a}}_{22} \end{gathered}$$

Simplify the equation $3{\mathrm{a}}_{23}=3{\mathrm{a}}_{23}$as follows.

$$\begin{gathered} 3{\mathrm{a}}_{23}=4{\mathrm{a}}_{23} \\ {\mathrm{a}}_{23}=0 \end{gathered}$$

Simplify the equation$3{\mathrm{a}}_{31}=2{\mathrm{a}}_{31}$as follows.

$$\begin{gathered} 3{\mathrm{a}}_{31}=2{\mathrm{a}}_{31} \\ {\mathrm{a}}_{31}=0 \end{gathered}$$

Simplify the equationrole="math" localid="1659848475865" $3{\mathrm{a}}_{32}=2{\mathrm{a}}_{32}$as follows.

$$\begin{gathered} 3{\mathrm{a}}_{32}=2{\mathrm{a}}_{32} \\ {\mathrm{a}}_{32}={\mathrm{a}}_{32} \end{gathered}$$

Simplify the equation$3{\mathrm{a}}_{33}=3{\mathrm{a}}_{33}$as follows.

$$\begin{gathered} 4{\mathrm{a}}_{33}=4{\mathrm{a}}_{33} \\ {\mathrm{a}}_{33}={\mathrm{a}}_{33} \end{gathered}$$

Substitute all the values of ${\mathrm{a}}_{\mathrm{ij}}$in the equation $\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]$as follows.

$$\begin{gathered} \mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right] \\ \mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& 0& 0\\ 0& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ 0& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right] \end{gathered}$$

Therefore, any matrix in the form$\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& 0& 0\\ 0& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ 0& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]$ satisfy the equation$\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}A=A\mathit{\left[}\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\mathit{\right]}$ .

Simplify the equation $\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& 0& 0\\ 0& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ 0& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]$as follows.

$$\begin{gathered} \mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& 0& 0\\ 0& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ 0& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right] \\ \mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& {\mathrm{a}}_{22}& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& {\mathrm{a}}_{23}\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& {\mathrm{a}}_{32}& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& {\mathrm{a}}_{33}\end{array}\right] \end{gathered}$$

$$\begin{gathered} \mathrm{A}={\mathrm{a}}_{11}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+{\mathrm{a}}_{22}\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]+{\mathrm{a}}_{23}\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+{\mathrm{a}}_{32}\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+{\mathrm{a}}_{33}\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right] \\ \mathrm{where}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\mathrm{and}\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right] \end{gathered}$$are linear independent.

Therefore, the matrix A is spanned by

$span\left\{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 1& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\right\}$.

Hence, the dimension of A is 5 and spanned by

$span\left\{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 1& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\right\}$.