Determinants and Cramer S Rule

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LESSON Reteach 4-4 Determinants and Cramer’s Rule A square matrix has the same number of rows as columns. The determinant of a square a b matrix is shown by . c d To find the determinant of a 2  2 matrix, find the product of each diagonal, beginning at the upper left corner. Then subtract. 轾a b a b det 犏 = =ad - cb 臌c d c d 轾2 3 2 3 det 犏 = =2(9) - 5(3) = 18 - 15 = 3 臌5 9 5 9 VerticalVertical bracketsbrackets indicateindicate aa determinant.determinant.

Find the determinant of each matrix. 轾-1 2 - 1 2 1. det 犏 = = -1(4) - ( - 5)(2) = 臌-5 4 - 5 4

3 1   3 1    2 4   2 4  3 1   1   1  2. det              1 1   1 1  2 4   2   4  2 4   2 4 

轾-3 - 4 - 3 - 4 3. det 犏 = = 3( ____ )  ( 1 ) ( ____ )  臌-1 - 6 - 1 - 6

轾-2.4 0.5 - 2.4 0.5 4. det 犏 = = 臌1.2 2 1.2 2

轾1 1 9 9 犏6 6 5. det 犏 = = 2 2 犏 -12 - 12 臌犏3 3

轾 2 2 8 8 犏 5 5 6. det 犏 = = 3 3 犏-15 - 15 臌犏 4 4

LESSON Reteach 4-4 Determinants and Cramer’s Rule (continued) x y  2 Use Cramer’s rule to solve a system of linear equations.  y7  2 x Step 1 Write the equations in standard form, ax  by  c. x  y  2 2x  y  7

Step 2 Write the coefficient matrix of the system of equations. Then find the determinant of the coefficient matrix.

轾a1 b 1 轾1 1 Coefficient matrix 犏 = 犏 Coefficient matrix 臌a2 b 2 臌2- 1

a b 1 1 DeterminantDeterminant D =1 1 = =1( - 1) - 2(1) = - 3 a2 b 2 2- 1 Step 3 Solve for x and y using Cramer’s rule. Remember to divide by the determinant.

TheThe coefficientscoefficients ofof xx inin thethe c1 b 1 2 1 coefficientcoefficient matrixmatrix areare replacedreplaced c b 7- 1 -2 - 7 - 9 x =2 2 = = = = 3 byby thethe constantconstant terms.terms. D -3 - 3 - 3

TheThe coefficientscoefficients ofof yy inin thethe a1 c 1 1 2 coefficient matrix are replaced coefficient matrix are replaced a2 c 2 2 7 7- 4 3 by the constant terms. y = = = = = -1 by the constant terms. D -3 - 3 - 3 The solution is (3, 1).

Use Cramer’s rule to solve each system of equations.  2x y   1 x y  1 7.  8.  4x y   5 3x 2 y  4 2 1 D = = D  4 1 -1 1 -5 1 = x  x = D 2- 1 4- 5 = y  y = D y x  3 3y 4 x  7 9.  10.  2x 2  y 9 6x  2 y D  x  x  y  y 