1. Warm up Solve this system using inverses: –1. x + y –z = -2 – 2x –y + z = 5 – -x + 2y + 2z = 1

2. Warm up Multiply: Find the inverse of the answer.

3. Lesson 11-4 Determinants Objective: To learn to find the determinant of a 2 x 2 and 3 x 3 matrix.

4. Determinant - a square array of numbers or variables enclosed between parallel vertical bars. ** To find a determinant you must have a SQUARE MATRIX!!** Finding a 2 x 2 determinant:

5. The Determinant of a Matrix The determinant of A, denoted by │ A │ or det(A) is defined as

6. Matrix A has an inverse if and only if The Determinant of a Matrix

7. Example 1 Find the determinant, tell whether the matrix has an inverse, and find the inverse (if it exists). det(A) = ad - bc = (2)(2) – (3)(1) = 1, so matrix A has an inverse AB = I 2(a) + 3(c) 1(a) + 2(c) 2(b) + 3(d) 1(b) + 2(d)

8. Find the determinant:

9. Practice 1) Find the determinant, and tell whether each matrix has an inverse. 2)

10. Finding a 3x3 determinant: Diagonal method Step 1: Rewrite first two columns of the matrix.

11. Step 2: multiply diagonals going up! -224+10+162= -52 Step 2: multiply diagonals going down! -126 +12 +240 =126 Step 3: Bottom minus top! 126 - (-52) 126 + 52 = 178

12. Step 2: multiply diagonals going up! -18+50+6= 38 Step 3: multiply diagonals going down! 45 - 15 + 8 = 38 Step 4: Bottom minus top! 38 - 38 = 0

13. Practice: Find the determinant: 1. 2. 3. 4. 5. 6.

14. Area of a Triangle using Determinants Given 3 points that form a triangle, area = ½ |det| Example: (-2, -4) (3, -9) (8, 4) Create a matrix with x’s in the 1 st column, y’s in the 2 nd column and 1’s in the third column

15. Det = 90 Area = ½ (90)= 45

16. Try: (3, -1) (7, 9) (-9, -7) Det = 96 Area = ½ (96) = 48