Question

Determine whether the statement is true or false and explain
a) There exist two 1*1 mattrices A and B such AB≠BA
b)There exist two 2*2 matrices A and B such that AB≠BA
c)There exist two nonzero 2*2 matrices A and B such that AB is the 2*2 zero matrix
d)There exist two nonzero 1*1 matrices A and A such that AB is the 2*2 zero matrix

Answer 1

a)FALSE
Let A=[x] and B=[y] be any two 1*1 matrices.
AB = [x]*[y] = [xy]
BA = [y]*[x] = [xy]
So AB=BA.

b)TRUE.
There are many such matrices. One example is:
$A=\left[\begin{array}{cc}1&2\\4&5\end{array}\right];\ B=\left[\begin{array}{cc}0&0\\0&1\end{array}\right] \\ \\AB= \left[\begin{array}{cc}0&2\\0&5\end{array}\right] ;\ BA= \left[\begin{array}{cc}0&0\\4&5\end{array}\right] \\ \\Thus\ AB \neq BA$

c)TRUE
You can find many such matrices.
$A=\left[\begin{array}{cc}3&6\\2&4\end{array}\right];\ B=\left[\begin{array}{cc}2&-8\\-1&4\end{array}\right]\\You\ can\ check\ that\ AB=0.$

d)FALSE.
Product of two 1*1 matrix is always a 1*1 matrix only. It can't be a 2*2 matrix.
If both A and B are non-zero, then AB can't be even a 1*1 zero matrix.