1. Algebra 2 Section 4-4 Matrices and Determinants

2. What You’ll Learn Why 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

3. 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. adgadg behbeh

4. 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. aeibfgcdh++

5. 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. gechfaidb++

6. 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. (aeibfgcdh)++ (gec hfa idb)+ + -

7. Example 1 Evaluate using diagonals.

8. Solution Example 1 First, rewrite the first two columns to the right of the determinant. 7 2 032032

9. 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

10. 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

11. 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) -15 + 0 + 112 97 2(3)(8)+ 2(4)(-1)+ 5(7)(0) 48 - 8 + 0 40 97 - 40 57

12. 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

13. Example 2 Find the area of the triangle whose vertices are located at (3,-4), (5,4), and (-3,2)

14. 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

15. Solution Example 2 3(4)(1) + (-4)(1)(-3) + (1)(5)(2) 34 12 + 12 + 10 -3(4)(1) + 2(1)(3) + (1)(5)(-4) -12 + 6 - 20 -26 34 - (-26)= 60 A = ½(60) A = 30 Square Units