Expert: Scott A Wilson - 10/21/2009

Question
Hi Scott, I have one question please help me regarding that i will very thankful to you. i.e.

Show that there exist no 2x2 matrices A and B such that AB-BA=I?

Thanks.

Answer
Let A be

a11   a12
a21   a22

and B be

b11   b12
b21   b22.

Now if we take AB we get
a11•b11 + a12•b21   a11•b12 + a12•b22
a21•b11 + a22•b21   a21•b12 + a22•b22

Also, we need to look at BA.  This gives
a11•b11 + a21•b12   a12•b11 + a22•b12
a11•b21 + a21•b22   a12•b21 + a22•b22

Now to compute I = AB - BA.
I11 = 1 = a12•b21 - a21•b12
I22 = 1 = a21•b12 - a12•b21

Notice that I11 = - I22.  For an identity matrix, this is not possible.