LA2.1.15

From Example Problems
Jump to: navigation, search

Without expanding the determinant prove that {\begin{vmatrix}1&bc&b+c\\a&ca&c+a\\1&ab&a+b\end{vmatrix}}={\begin{vmatrix}1&a&a^{2}\\1&b&b^{2}\\1&c&c^{2}\end{vmatrix}}\,

LHS={\begin{vmatrix}1&bc&b+c\\a&ca&c+a\\1&ab&a+b\end{vmatrix}}\,

By C-3-(a+b+c)C_{1}\, we get

{\begin{vmatrix}1&bc&-a\\1&ca&-b\\1&ab&-c\end{vmatrix}}\,

By aR_{1},bR_{2},cR_{3}\,

{\frac  {1}{abc}}{\begin{vmatrix}a&abc&-a^{2}\\b&abc&-b^{2}\\c&abc&-c^{2}\end{vmatrix}}\,

{\frac  {abc}{abc}}{\begin{vmatrix}a&1&-a^{2}\\b&1&-b^{2}\\c&1&-c^{2}\end{vmatrix}}\,

{\begin{vmatrix}1&a&-a^{2}\\1&b&b^{2}\\1&c&c^{2}\end{vmatrix}}\,=RHS


Main Page:Linear Algebra