<p>Section 8.5: Guided Notes Applications of Matrices and Determinants</p><p>I. Area of a Triangle </p><p> The area of the triangle with vertices (x 1 , y 1 ), (x 2 , y 2 ), and </p><p>(x 3 , y 3 ) is</p><p> x y 1 1 1 1 Area = ± x y 1 2 2 2 x3 y 3 1</p><p>Example 1. Find the area of the triangle with vertices (1, -1), (2, 3), and (5, -3).</p><p>II. Collinear Points</p><p> If the area in the above formula equals zero, then the three points must be collinear. </p><p> Test for collinear points: the three points (x 1 , y 1 ), (x 2 , y 2 ), and </p><p>(x 3 , y 3 ) are collinear if and only if</p><p>Example 2. Determine whether the points (0, 1), (4, 4), and (8, 7) are collinear. III. Cramer’s Rule </p><p> If we solve:</p><p> by elimination, we obtain</p><p> and</p><p>Tip: Try to see the pattern. All denominators are the determinant of the coefficient matrix, the numerator of x is the same as the denominator except with the x-coefficients replace by the constants, and the numerator of y is the same as the denominator except with the y-coefficients replace by the constants.</p><p>Example 3. Use Cramer’s Rule solve the system of linear equations. Generalized Cramer’s Rule:</p><p> If a system of n linear equations with n variables has a coefficient matrix A with a nonzero determinant, the solutions of the system are of the form</p><p> where A i is formed from A deleting the ith column and replacing it with the column of constants.</p><p>Example 4. Use Cramer’s Rule solve the system of linear equations. IV. Cryptography </p><p> We are going to use matrices to encode and decode messages. We start by assigning each letter in the alphabet a number, starting with 0 being a space, 1 is A, and so on to 26 is Z.</p><p>A=1;B=2;C=3;D=4;E=5;F=6;G=7;H=8;I=9;J=10;K=11;L=12;M=13; N=14;O=15;P=16;Q=17;R=18;S=19;T=20;U=21;V=22;W=23;X=24; Y=25;Z=26; SPACE=0</p><p>Example 5. Use the matrix</p><p> to decode the message. The encoded message is: 59 41 –38 54 25 –54 15 21 5 18 18 –5 41 29 –32 41 19 –41</p><p> Find A -1  Transform the encoded message into encoded matrices (1 3)  Multiply the encoded matrices by A -1  Decode using the letter-number assignments</p>