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Thus, to prevent pivoting, $b = 0$ must be the case also. +$a \neq 0$ then $b \neq 0$ also, as $\frac{b}{a}$ will not be defined + +\newpage +\section{Question 2} +\hr +For A to be invertible, $det(A) \neq 0$. + +$A = \begin{vmatrix} + 1 & 1 & 1 & 1 \\ + 1 & 2 &-1 & c \\ + 1 & 4 & 1 & c^2 \\ + 1 & 8 &-1 & c^3 \\ +\end{vmatrix} +$ + +\begin{align*} + det(A) + & = (1)\begin{vmatrix} + 2 &-1 & c \\ + 4 & 1 & c^2 \\ + 8 &-1 & c^3 \\ + \end{vmatrix} + + (-1)\begin{vmatrix} + 1 &-1 & c \\ + 1 & 1 & c^2 \\ + 1 &-1 & c^3 \\ +\end{vmatrix} + +(1)\begin{vmatrix} + 1 & 2 & c \\ + 1 & 4 & c^2 \\ + 1 & 8 & c^3 \\ +\end{vmatrix} + +(-1)\begin{vmatrix} + 1 & 2 &-1 \\ + 1 & 4 & 1 \\ + 1 & 8 &-1 \\ +\end{vmatrix} \\ + & = (1)\left((2)\begin{vmatrix} + 1 & c^2 \\ + -1 & c^3 \\ + \end{vmatrix} + + (1)\begin{vmatrix} + 4 & c^2 \\ + 8 & c^3 \\ + \end{vmatrix} + + (c)\begin{vmatrix} + 4 & 1 \\ + 8 &-1 \\ + \end{vmatrix}\right)\\ + & + (-1)\left((1)\begin{vmatrix} + 1 & c^2 \\ + -1 & c^3 \\ + \end{vmatrix} + + (1)\begin{vmatrix} + 1 & c^2 \\ + 1 & c^3 \\ + \end{vmatrix} + + (c)\begin{vmatrix} + 1 & 1 \\ + 1 &-1 \\ + \end{vmatrix}\right)\\ + & + (1)\left((1)\begin{vmatrix} + 4 & c^2 \\ + 8 & c^3 \\ + \end{vmatrix} + + (-2)\begin{vmatrix} + 1 & c^2 \\ + 1 & c^3 \\ + \end{vmatrix} + + (c)\begin{vmatrix} + 1 & 4 \\ + 1 & 8 \\ + \end{vmatrix}\right)\\ + & + (-1)\left((1)\begin{vmatrix} + 4 & 1 \\ + 8 &-1 \\ + \end{vmatrix} + + (-2)\begin{vmatrix} + 1 & 1 \\ + 1 &-1 \\ + \end{vmatrix} + + (-1)\begin{vmatrix} + 1 & 4 \\ + 1 & 8 \\ + \end{vmatrix}\right)\\ + & = (2)(c^3+c^2) + (4c^3-8c^2) + (c)(-4-8)\\ + & + (-1)((c^3+c^2) + (c^3-c^2) + (c)(-1 - 1))\\ + & + (4c^3-8c^2) + (-2)(c^3-c^2) + (c)(8-4)\\ + & + (-1)((-4-8) + (-2)(-1-1) + (-1)(8-4)\\ + & = 6c^3 - 12c^2 - 6c + 12 \\ + & = 6(c - 2)(c - 1)(c + 1) +\end{align*} + +Therefore, $det(A) = (c-2)(c-1)(c+1) \neq 0$, \\ +$c \neq 2, c \neq 1, c \neq -1$ + +Actually, this question can be solved by observation. The determinant will be 0 if 2 columns are a multiple of one another. By observation, we can see that if $c = 1$, then the 4th and 1st column will be the same. If $c = 2$, then 2,4 and $c=-1$, then 3,4 will be the same. + +% ------------------------------------------------------------------------------ + +\end{document}