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Chapter 6: Problem 54
What is a cryptogram?
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Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices\(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) is $$ \text { Area }=\pm \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right| $$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use this information to work. Use determinants to find the area of the triangle whose vertices are \((3,-5),(2,6),\) and \((-3,5)\).
Use Cramer's rule to solve each system. $$ \begin{aligned}&3 x+2 z=4\\\&5 x-y=-4\\\&4 y+3 z=22\end{aligned} $$
Evaluate each determinant. $$ \left|\begin{array}{rrr}4 & 0 & 0 \\\3 & -1 & 4 \\\2 & -3 & 5\end{array}\right| $$
In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ 5 C-2 B $$
Use a graphing utility to evaluate the determinant for the given matrix. $$ \left[\begin{array}{rrrr}3 & -2 & -1 & 4 \\\\-5 & 1 & 2 & 7 \\\2 & 4 & 5 & 0 \\\\-1 & 3 & -6 & 5\end{array}\right] $$
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