8.3 Determinants

Evaluate:

(a) [math]\begin{vmatrix}1&0\\3&7\end{vmatrix}[/math]

(b) [math]\begin{vmatrix}2&-9\\3&8\end{vmatrix}[/math]

(c) [math]\begin{vmatrix}1&-2&2\\3&-13&4\\9&0&9\end{vmatrix}[/math]

(d) [math]\begin{vmatrix}9&98&99\\0&8&97\\0&0&7\end{vmatrix}[/math]

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]

Consider a matrix [math]A=\begin{bmatrix}a&-b\\c&-a\end{bmatrix}[/math] where [math]a, b[/math], and [math]c[/math] are real numbers. If [math]\det(A)=2[/math], what is the value of [math]\det(A+A^{-1})[/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.

Evaluate:

(a) [math]\begin{vmatrix}-5&4\\-3&8\end{vmatrix}[/math]

(b) [math]\begin{vmatrix}1&0&2\\3&5&9\\-2&-3&0\end{vmatrix}[/math]

(c) [math]\begin{vmatrix}-4&3&-2\\3&0&3\\4&7&1\end{vmatrix}[/math]

Find [math]x[/math] such that [math]\begin{bmatrix}1-x&1\\6&2-x\end{bmatrix}[/math] is a singular matrix. If you have two or more answers, write the smallest one.

(Calculator) Let [math]A=\begin{bmatrix}1&-9&7&-1\\0&8&-4&-7\\1&4&5&0\\-3&-9&2&6\end{bmatrix}[/math]. Evaluate [math]C_{21}[/math].

Let [math]A(2,-1), B(4,0), C(-3,5)[/math] be three points on a plane. Find the area of the triangle [math]ABC[/math].