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In this lesson, you will learn how to evaluate the determinant of a 3x3 matrix using diagonals. The process involves rewriting the first two columns next to the original matrix and drawing diagonals to calculate products. You'll also discover how to find the area of a triangle given its vertices using determinants. This technique showcases the practical applications of matrices and determinants in solving geometry and geography problems, enabling you to harness these concepts effectively for various mathematical challenges.
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Algebra 2 Section 4-4 Matrices and Determinants
What You’ll LearnWhy It’s Important • To evaluate the determinant of a 3 x 3 matrix, • To find the area of a triangle, given the coordinates of its vertices • You can use matrices and determinants to solve problems involving geometry and geography
Evaluating a third-order determinant using Diagonals • In this method, you begin by writing the first two columns on the right side of the determinant. a d g b e h
Evaluating a third-order determinant using Diagonals • Next, draw diagonals from each element of the top row of the determinant downward to the right. Find the product of the elements on each diagonal. aei + bfg + cdh
Evaluating a third-order determinant using Diagonals • Then, draw diagonals from the elements in the third row of the determinant upward to the right. Find the product of the elements on each diagonal. gec + hfa + idb
(aei + bfg + cdh) + (gec + idb) hfa Evaluating a third-order determinant using Diagonals • To find the value of the determinant, add the products of the first set of diagonals and then subtract the sum of the products of the second set of diagonals. -
Example 1 • Evaluate using diagonals.
Solution Example 1 • First, rewrite the first two columns to the right of the determinant. -1 7 2 0 3 2
Solution Example 1 • Next, find the products of the elements of the diagonals (going down). -1(3)(5) + 0(4)(2) + 8(7)(2) -15 + 0+ 112 97
Solution Example 1 • Next, find the products of the elements of the diagonals (going up). 2(3)(8) + 2(4)(-1) + 5(7)(0) 48- 8 + 0 40
Solution Example 1 • To find the value of the determinant, add the products of the first set of diagonals (going down) and then subtract the sum of the products of the second set of diagonals (going up). -1(3)(5) + 0(4)(2) + 8(7)(2) 2(3)(8) + 2(4)(-1) + 5(7)(0) -15 + 0+ 112 48- 8 + 0 97 40 97 - 40 57
What is the purpose of determinants? • One very powerful application of determinants is finding the areas of polygons. • The formula below shows how determinants serve as a mathematical tool to find the area of a triangle when the coordinates of the three vertices are given. Area of Triangles The area of a triangle having vertices at (a,b), (c,d), and (e,f) is |A|, where Notice that it is necessary to use the absolute value of A to guarantee a nonnegative value for area
Example 2 • Find the area of the triangle whose vertices are located at (3,-4), (5,4), and (-3,2)
Solution Example 2 • Find the area of the triangle whose vertices are located at (3,-4), (5,4), and (-3,2) • Assign values to a, b, c, d, e, and f and substitute them into the area formula and evaluate. • (a,b) = (3,-4) • (c,d) = (5,4) • (e,f) = (-3,2) Evaluate the determinant and then multiply by ½
Solution Example 2 3(4)(1) + (-4)(1)(-3) + (1)(5)(2) 12 +12+ 10 34 -3(4)(1) + 2(1)(3) + (1)(5)(-4) A = ½(60) -12+6- 20 -26 A = 30 Square Units 34 - (-26) = 60
