Let $S=\left\{\left(\begin{array}{cc}\hfill -1\hfill & \hfill a\hfill \\ \hfill 0\hfill & \hfill b\hfill \end{array}\right)a,b\in \left\{1,2,3,...100\right\}\right\};$ and let Tn = $\left\{A\in S:A^{n\left(n+1\right)}=I\right\}.$ Then the number of elements in ${\displaystyle \underset{n=1}{\overset{100}{\cap }}{T}_n}is............$

<p>&nbsp;<math><mrow><mi>S</mi><mo>=</mo><mrow><mo> {</mo><mrow><mrow><mo> (</mo><mrow><mtable><mtr columnalign="center"><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mi>a</mi></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mi>b</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow></mrow><mo>, </mo><mi>a</mi><mo>, </mo><mi>b</mi><mo>∈</mo><mn>1</mn><mo>, </mo><mn>2</mn><mo>, </mo><mn>3</mn><mo>, </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mn>1</mn><mn>0</mn><mn>0</mn></mrow></math></p><p><math><mrow><mo>∴</mo><mi>A</mi><mo>=</mo><mrow><mo> (</mo><mrow><mtable><mtr columnalign="center"><mtd columnalign="center"><mo>−</mo><mn>1</mn></mtd><mtd columnalign="center"><mi>a</mi></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mi>b</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow></math>&nbsp;then even power of A as A =&nbsp;<math><mrow><mrow><mo> (</mo><mrow><mtable><mtr columnalign="center"><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mn>0</mn></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>1</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math></p><p>If b = 1 &amp;&nbsp;<math><mrow><mi>a</mi><mo>∈</mo><mrow><mo> {</mo><mrow><mn>1</mn><mo>, </mo><mn>2</mn><mo>, </mo><mn>3</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mn>1</mn><mn>0</mn><mn>0</mn></mrow><mo>}</mo></mrow></mrow></math>&nbsp;and n (n + 1) is always even</p><p><math><mrow><mo>∴</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>, </mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>, </mo><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>, </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>, </mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math>&nbsp;are all 1 for b = 1 and each value of a.</p><p><math><mrow><mo>∴</mo><mstyle displaystyle="true"><munderover><mo>∩</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow></munderover><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn></mrow></mstyle></mrow></math></p>

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