Determinants Determinant a square array of numbers or
Determinants
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:
Find the determinant: (-5)(8) – (-7)(11) (-40) – (-77) 37 (3)(5) – (2)(-1) (15) – (-2) 17
Challenge: (-4)(x) – (1)(3) = 17 -4 x – 3 = 17 -4 x = 20 x = -5
3 x 3 Determinant: Diagonal method Step 1: Rewrite first two columns of the matrix.
-224 +10 +162 = -52 Step 2: multiply diagonals going down! Step 3: multiply diagonals going up! -126 +12 +240 =126 - (-52) = 178 Last Step: (step 2) – (step 3)
Now you try: -18 +50 +6 = 38 38 - 38 =0 45 - 15 + 8 = 38
3 x 3 Determinant: Expansion by Minors method Imagine crossing out the first row. Now take the double-crossed element. . . And the first column. And multiply it by the determinant of the remaining 2 x 2 matrix 3 8 ¼ 2 0 -¾ 4 180 11
3 x 3 Determinant: Expansion by Minors method Keep first row crossed out Now cross out the second column 3 8 ¼ 2 0 -¾ 4 180 11 Now take the negative of the double-crossed element. . . And multiply it by the determinant of the remaining 2 x 2 matrix
3 x 3 Determinant: Expansion by Minors method Finally, cross out first row and last column 3 2 4 Use double-crossed element And multiply it by the determinant of the remaining 2 x 2 matrix o t s l a t o t b u s 0 e 9 e r + h t 0 l 2 0 dd a-¾ + A 405 295 : r e geth 8 ¼ 180 11
Now you try: : r e h oget t s l a t o t b u s 6 e e 2 r h + t l 1 l 2 a d + d A -5 0 5[9 -10] - (-1) [-6 -15] 5(-1) 1 (-21) -5 -21 2[4 - (-9)] 2(13) 26
Use determinant to find area of a triangle Find the area of a triangle whose vertices are located at (– 3, – 3), (– 1, 2), and (3, – 1). A= (a, b) = (– 3, – 3) (c, d) = (– 1, 2) (e, f) = (3, – 1) = = [– 26] or – 13 13 units 2 Simplify. *Remember area cannot be negative, thus we take the absolute value
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