Question

Show that if A is a$2\times 2$ matrix such that AB=BA whenever B is an$2\times 2$ matrix , then $A=cI$, where C is a real number and I is the$2\times 2$ identity matrix.

Short answer

$A=\left[\begin{array}{l}a\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}a\end{array}\right]=aI$

Step-by-step solution

Step: 1

Suppose that $A=\left[\begin{array}{l}a\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}b\\ c\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}d\end{array}\right]$

Let$B=\left[\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\end{array}\right]$

Step: 2

Because AB=BA, if follows that c=0 and a=d.

Hence$A=\left[\begin{array}{l}a\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}a\end{array}\right]=aI$