If a, b and c are integers, then number of matrices $\mathrm{A}=\left[\begin{array}{l}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{c}\\ \mathrm{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{c}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}\\ \mathrm{c}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}\end{array}\right]$ which are possible such that AA^T = I is ___________.

$\begin{array}{l}{\mathrm{AA}}^{\mathrm{T}}=\mathrm{I}\\ \therefore {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=1\text{ and }\mathrm{ab}+\mathrm{bc}+\mathrm{ca}=0\end{array}$

This is possible when

$$\begin{gathered} \text{ (i) }\mathrm{a}=\pm 1,\mathrm{b}=\mathrm{c}=0 \\ \text{ (ii) }\mathrm{a}=0,\mathrm{b}=\pm 1,\mathrm{c}=0 \\ \text{ (iii) }\mathrm{a}=0,\mathrm{b}=0,\mathrm{c}=\pm 1 \end{gathered}$$

So, there are six such matrices A.