This video may be helpful. Matrix Multiplication video

a)

$$\begin{eqnarray*} AB &=& \begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}\begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \simplify[]{{a11}{b11}+{a12}{b21}}&\simplify[]{{a11}{b12}+{a12}{b22}}\\ \simplify[]{{a21}{b11}+{a22}{b21}}&\simplify[]{{a21}{b12}+{a22}{b22}}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \var{ab11}&\var{ab12}\\ \var{ab21}&\var{ab22}\\ \end{pmatrix} \end{eqnarray*}$$

b)

$$\begin{eqnarray*} BA &=& \begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix}\begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \simplify[]{{b11}{a11}+{b12}{a21}}&\simplify[]{{b11}{a12}+{b12}{a22}}\\ \simplify[]{{b21}{a11}+{b22}{a21}}&\simplify[]{{b21}{a12}+{b22}{a22}}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \var{ba11}&\var{ba12}\\ \var{ba21}&\var{ba22}\\ \end{pmatrix} \end{eqnarray*}$$

c)

$$\begin{eqnarray*} CB &=& \begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix}\begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \simplify[]{{c11}{b11}+{c12}{b21}}&\simplify[]{{c11}{b12}+{c12}{b22}}\\ \simplify[]{{c21}{b11}+{c22}{b21}}&\simplify[]{{c21}{b12}+{a22}{b22}}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \var{cb11}&\var{cb12}\\ \var{cb21}&\var{cb22}\\ \end{pmatrix} \end{eqnarray*}$$

d)

$$\begin{eqnarray*} AC &=& \begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}\begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \simplify[]{{a11}{c11}+{a12}{c21}}&\simplify[]{{a11}{c12}+{a12}{c22}}\\ \simplify[]{{a21}{c11}+{a22}{c21}}&\simplify[]{{a21}{c12}+{a22}{c22}}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \var{ac11}&\var{ac12}\\ \var{ac21}&\var{ac22}\\ \end{pmatrix} \end{eqnarray*}$$