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Warm up

Warm up. Solve this system using inverses: 1. x + y –z = -2 2x –y + z = 5 -x + 2y + 2z = 1. Warm up. Multiply: Find the inverse of the answer. Lesson 11-4 Determinants. Objective: To learn to find the determinant of a 2 x 2 and 3 x 3 matrix.

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Warm up

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

  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) 2(b) + 3(d) 1(a) + 2(c) 1(b) + 2(d)

  8. Find the determinant:

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

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

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

  12. -18 +50 +6 = 38 45 - 15 + 8 Step 2: multiply diagonals going up! Step 3: multiply diagonals going down! = 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 1st column, y’s in the 2nd 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

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