91Ó°ÊÓ

Consider the matrix $B=\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$such that $\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]A=A\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$where

localid="1659755052566" $\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]$

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 in the equation$\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]A=A\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$as

follows.

localid="1659755291081" $$\begin{gathered} \left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]A=A\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right] \\ \left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]=\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right] \\ \left[\begin{array}{ccc}2{\mathrm{a}}_{11}& 2{\mathrm{a}}_{12}& 2{\mathrm{a}}_{13}\\ 3{\mathrm{a}}_{21}& 3{\mathrm{a}}_{22}& 3{\mathrm{a}}_{23}\\ 4{\mathrm{a}}_{31}& 4{\mathrm{a}}_{32}& 4{\mathrm{a}}_{33}\end{array}\right]=\left[\begin{array}{ccc}2{\mathrm{a}}_{11}& 3{\mathrm{a}}_{12}& 4{\mathrm{a}}_{13}\\ 2{\mathrm{a}}_{21}& 3{\mathrm{a}}_{22}& 2{\mathrm{a}}_{23}\\ 2{\mathrm{a}}_{31}& 3{\mathrm{a}}_{32}& 4{\mathrm{a}}_{33}\end{array}\right] \end{gathered}$$

Compare the equation both side as follows.

$$\begin{gathered} 2{\mathrm{a}}_{11}=2{\mathrm{a}}_{11} \\ 2{\mathrm{a}}_{12}=3{\mathrm{a}}_{12} \\ 2{\mathrm{a}}_{13}=4{\mathrm{a}}_{13} \\ 3{\mathrm{a}}_{21}=2{\mathrm{a}}_{21} \end{gathered}$$

Further, compare the equation as follows.

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

Further, compare the equation as follows.

$4a_{33}=4a_{33}$

Simplify the equation $2{\mathrm{a}}_{11}=2{\mathrm{a}}_{11}$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.

$$\begin{gathered} 2{\mathrm{a}}_{12}=3{\mathrm{a}}_{12} \\ {\mathrm{a}}_{12}=0 \end{gathered}$$

Simplify the equation$2{\mathrm{a}}_{13}=4{\mathrm{a}}_{13}$as follows.

$$\begin{gathered} 2{\mathrm{a}}_{13}=4{\mathrm{a}}_{13} \\ {\mathrm{a}}_{13}=0 \end{gathered}$$

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 equationlocalid="1659755751369" $3{\mathrm{a}}_{22}=3{\mathrm{a}}_{22}$as follows.

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

Simplify the equation $3{\mathrm{a}}_{23}=4{\mathrm{a}}_{23}$as follows.

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

Simplify the equation$4{\mathrm{a}}_{31}=2{\mathrm{a}}_{31}$as follows.

$$\begin{gathered} 4{\mathrm{a}}_{31}=2{\mathrm{a}}_{31} \\ {\mathrm{a}}_{31}=0 \end{gathered}$$

Simplify the equation$4{\mathrm{a}}_{32}=3{\mathrm{a}}_{32}$as follows.

$$\begin{gathered} 4{\mathrm{a}}_{32}=3{\mathrm{a}}_{32} \\ {\mathrm{a}}_{32}=0 \end{gathered}$$

Simplify the equation$4{\mathrm{a}}_{33}=4{\mathrm{a}}_{33}$as follows.

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

Substitute the values 0 for ${\mathrm{a}}_{\mathrm{ij}}$a where $i≠j$and 1 for ${\mathrm{a}}_{\mathrm{ii}}$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.

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

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

$\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]A=A\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$

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

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

Therefore, the matrix Ais spanned by span $\left\{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\right\}$.

Hence, the dimension of A is 3 and spanned by span$\left\{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\right\}$ .