Question
Question: Show that \[x = 2\] is a root of the equation \[\left| {\begin{array}{*{20}{c}} x&{ - 6}&{ - 1} ...
Show that x=2 is a root of the equation \left| {\begin{array}{*{20}{c}} x&{ - 6}&{ - 1} \\\ 2&{ - 3x}&{x - 3} \\\ { - 3}&{2x}&{2 + x} \end{array}} \right| = 0
Explanation
Solution
- Roots of an equation axn+bxn−1+........c=0are those values of x which give the value of the equation equal to zero when substituted in the equation. Let y be a root of the equation, we can write (x−y)=0is a factor of the equation. On equating the factor we write x=yis a root of the equation.
- Row transformation is a way of transforming elements of a row using operations like addition, multiplication etc with respect to any other row of the matrix.
- Determinant of a matrix \left[ {\begin{array}{*{20}{c}} a&b;&c; \\\ d&e;&f; \\\ g&h;&i; \end{array}} \right] = a(ei - hf) - b(di - fg) + c(dh - eg)
Complete step-by-step answer:
We are given the determinant \left| {\begin{array}{*{20}{c}}
x&{ - 6}&{ - 1} \\\
2&{ - 3x}&{x - 3} \\\
{ - 3}&{2x}&{2 + x}
\end{array}} \right|...............… (1)
We apply row transformations in order to make the terms of determinant simple.
Apply R3→R3−R1