Algebra Archive: Questions from July 22, 2023
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Example: Verify the following. det 1 x x² 1 y y² 1 Z 22 e-x xex xe-x e-x 1 x x2 = 1 y y² = (x-2)(x - y)(z- y). 1 Z N # 0. N
Example: Verify the following. \[ \left|\begin{array}{cc} e^{-x} & x e^{-x} \\ -x e^{-x} & e^{-x} \end{array}\right| \neq 0 \] \[ \operatorname{det}\left(\left[\begin{array}{lll} 1 & x & x^{2} \\ 1 &2 answers -
2. Find the preimage of \( w \) if \( T(v)=\left(v_{1}+v_{2}, v_{2}+v_{3}, v_{3}\right), w=(0,1,-2) \).2 answers -
varify the following
\( \operatorname{det}\left(\left[\begin{array}{lll}1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2}\end{array}\right]\right)=\left|\begin{array}{lll}1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2}\end{2 answers -
Example: Verify the following. \[ \mathbf{f}=\left|\begin{array}{lll} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array}\right|=(x-z)(x-y)(z-y) \]2 answers -
Gra ph. y y 1X2-1 -1/2X-2
\( \operatorname{Grs} p\left\{\begin{array}{l}y \leqslant \frac{1}{4} x^{2}-1 \\ y>-\frac{1}{2} x-2\end{array}\right. \)2 answers -
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