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Let us take two matrices A and X of order 2 x 2 where such that

$A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)andX=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)$$A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)andX=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)$

Then the matrix AX and XA are of order 2 x 2 as well.

We have

$A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)andX=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)$

$$\begin{gathered} AX=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right) \\ =\left(\begin{array}{cc}{a}_{11}{b}_{11}+a_{12}{b}_{12}& a_{11}{b}_{12}+a_{12}{b}_{22}\\ a_{21}{b}_{11}+a_{22}{b}_{21}& a_{21}{b}_{12}+a_{22}{b}_{22}\end{array}\right) \end{gathered}$$

$$\begin{gathered} wehave \\ A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)andX=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right) \end{gathered}$$

role="math" localid="1659347960061" $$\begin{gathered} XA=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right) \\ =\left(\begin{array}{cc}{a}_{11}{b}_{11}+a_{21}{b}_{12}& a_{12}{b}_{11}+a_{22}{b}_{12}\\ a_{11}{b}_{21}+b_{22}{b}_{21}& b_{21}{b}_{12}+b_{22}{b}_{22}\end{array}\right) \end{gathered}$$

We have proved AX = XA.

Since the two matrices are equal if (and only if) their corresponding elements are equal.

$$\begin{gathered} a_{11}{b}_{11}+a_{12}{b}_{21}=a_{11}{b}_{11}+a_{21}{b}_{12} \\ {a}_{11}{b}_{12}+b_{21}{a}_{12}=a_{12}{b}_{11}+a_{22}{b}_{12} \\ {a}_{21}{b}_{11}+a_{22}{b}_{21}=a_{11}{b}_{21}+a_{21}{b}_{22} \\ {a}_{21}{b}_{12}+a_{22}{b}_{22}=a_{12}{b}_{21}+a_{22}{b}_{22} \end{gathered}$$

Find the value of X.

$X=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ \frac{b_{12}{a}_{22}}{a_{21}}& \frac{a_{21}{b}_{11}+\left(a_{12}-a_{11}\right)}{a_{21}}{b}_{12}\end{array}\right)$

All 2 脳 2 matrices X such that AX = X A for all 2 脳 2 matrices A is

$X=\left(\begin{array}{cc}{b}_{11}& b_{12}\\ \frac{b_{12}{a}_{22}}{a_{21}}& \frac{a_{21}{b}_{11}+\left(a_{12}-a_{11}\right)}{a_{21}}{b}_{12}\end{array}\right)$.