Question

Using properties of determinants, prove that : $\begin{vmatrix}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{vmatrix}$ = $a b c+b c+c a+a b.$

Using properties of determinants, prove that : $\begin{vmatrix}a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b\end{vmatrix}$ = $2(a+b+c)^{3}.$

Using properties of determinants, prove the following : $\begin{vmatrix}a & a² & bc \\ b & b² & ca \\ c & c² & ab\end{vmatrix}$ = $(a-b)(b-c)(c-a)(b c+c a+a b).$

Using properties of determinants, prove that : $\begin{vmatrix}2y & y-z-x & 2y \\ 2z & 2z & z-x-y \\ x-y-z & 2x & 2x\end{vmatrix}$ = $(x+y+z)^{3}.$

Using properties of determinants, prove that : $\begin{vmatrix}x²+1 & xy & xz \\ xy & y²+1 & yz \\ xz & yz & z²+1\end{vmatrix}$

Using properties of determinants, solve for x : $\begin{vmatrix}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{vmatrix}$ = 0

Using properties of determinants, prove that $\begin{vmatrix}x+y & x & x \\ 5x+4y & 4x & 2x \\ 10x+8y & 8x & 3x\end{vmatrix}$ = x³