4.3 Determinants & Cramer’s Rule

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Presentation transcript:

4.3 Determinants & Cramer’s Rule OBJ: To Evaluate determinants of 2 x 2 and 3 x 3 matrices & Use Cramer's rule to solve systems of linear equations 4.3 Determinants & Cramer’s Rule Determinant of a Matrix A is denoted by: detA or |A| Determinant of a 2x2: the difference of the product of the diagonals = ad - cb

Ex: Find the determinant of = 7(3) - 2(2) = 21 - 4 = 17 Practice: =1(7) - 2(4) det = = 7 - 8 = -1

Determinant of a 3x3:. recopy the 1st two columns, then subtract the Determinant of a 3x3: recopy the 1st two columns, then subtract the sum of the product of the diagonals Ex: -3 -2 0 1 2 = (2)(0)(6) + (-3)(3)(1) + (4)(-2)(2) - (1)(0)(4) + (2)(3)(2) + (6)(-2)(-3) = (0 + -9 + -16) – (0 + 12 + 36) = -25 - 48 = -73

Cramer’s Rule Let A be the coefficient matrix ****You can use determinants to solve systems of equations: Cramer’s Rule Let A be the coefficient matrix Linear System Coeff Matrix ax+by=e cx+dy=f If detA 0, then the system has exactly one solution: and

Ex: Solve the system (-1, 2) 8x+5y=2 2x-4y=-10 The coefficient matrix is: and and So: (-1, 2)

Practice: Solve the system 2x+y=1 3x-2y=-23 The solution is: (-3,7)