question_answer

If  then show that A satisfies the following equation.

\[{{A}^{3}}-4{{A}^{2}}+11I-3A=O\]

OR

If \[A+B+C=\pi ,\]show that

\[=-\sin (A\,-B)sin(B\,-C)sin(C\,-A).\]

Answer:

We have,

\[\therefore \]

Now,

?(i)

\[\therefore \]

?(ii)

Now,     ?(iii)

\[\therefore \]\[{{A}^{3}}-4{{A}^{2}}+11/3A=0\]

[from Eqs. (i), (ii) and (iii)]

Hence proved.

OR

The given determinant

\[[{{C}_{1}}\to {{C}_{1}}+{{C}_{3}}]\]

\[[{{R}_{2}}\to {{R}_{2}}-{{R}_{1}},\,\,{{R}_{3}}-{{R}_{1}}]\]

[taking \[2\sin (A-B)\]and\[2\sin (A-C)\]common from\[{{R}_{2}}\]and \[{{R}_{3}}\] respectively]

\[[\because A+B+C=\pi ]\]

\[=\sin \,(A-B)\sin \,(A-C)[\sin B\cos C-\cos B\sin C]\]

\[=\sin \,(A-B)\sin \,(A-C)\sin \,(B-C)\]

\[=-\sin \,(A-B)\sin \,(B-C)\sin \,(C-A)\]

Hence proved.