8.3 Determinants (in class)

Let [math]\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=1[/math]. Evaluate:

(a) [math]\begin{vmatrix}c&2b&a\\f&2e&d\\3i&6h&3g\end{vmatrix}[/math]

(b) [math]\begin{vmatrix}g&a&d\\h&b&e\\i&c&f\end{vmatrix}[/math]

(c) [math]\begin{vmatrix}a+b&b+c&c+a\\d+e&e+f&f+d\\g+h&h+i&i+g\end{vmatrix}[/math]

What is the coefficient of [math]x^2[/math] of the determinant [math]\begin{vmatrix}x+1&-17&32\\61&2x+1&-9\\10&23&3x+1\end{vmatrix}[/math]?

Let [math]A=\begin{vmatrix}1&2&0&0\\4&3&0&0\\0&0&5&6\\0&0&8&7\end{vmatrix}[/math]. If [math]C_{ij}[/math] is the cofactor of [math]A[/math], what is [math]C_{11}+C_{22}+C_{33}+C_{44}[/math]?

Let [math]a, b, c\in\mathbb{R}[/math]. Use the identity [math]\begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix}=(a-b)(b-c)(c-a)[/math] to evaluate:

(a) [math]\dfrac{a}{(a-b)(a-c)}+\dfrac{b}{(b-c)(b-a)}+\dfrac{c}{(c-a)(c-b)}[/math]

(b) [math]\dfrac{a^2}{(a-b)(a-c)}+\dfrac{b^2}{(b-c)(b-a)}+\dfrac{c^2}{(c-a)(c-b)}[/math]

Note: The determinant is called the Vandermonde determinant.