Answer :
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:
[tex] 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[/tex]
c)TRUE
You can find many such matrices.
[tex] 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.[/tex]
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.
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:
[tex] 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[/tex]
c)TRUE
You can find many such matrices.
[tex] 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.[/tex]
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.