4.3 Determinants & Cramer’s Rule

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Presentation transcript:

4.3 Determinants & Cramer’s Rule

Objectives/Assignment

Warm-Up (2,1) Solve the system of equations: What is the product of these matrices?

Associated with each square matrix is a real number called it’s determinant. We write The Determinant of matrix A as det A or |A|

Here’s how to find the determinant of a square 2 x 2 matrix: Multiply Multiply 24 (1st) Now subtract these two numbers. - -16 is the determinant of this matrix 24 (1st) 40 (2nd ) = -16

In General

Determinant of a 3 x 3 Matrix (gec +hfa +idb) Now Subtract the 2nd set products from the 1st. a b d e (aei + bfg + cdh) - (gec + hfa + idb) g h (aei+ bfg +cdh)

Compute the Determinant of this 3 x 3 Matrix (0 +4 +8) Now Subtract the 2nd set products from the 1st. 2 -1 -2 0 =-25 (-13) - (12) 1 2 (0+ -1 -12)

You can use a determinant to find the Area of a Triangle (a,b) The Area of a triangle with verticies (a,b), (c,d) and (e,f) is given by: (e,f) (c,d) Where the plus or minus sign indicates that the appropriate sign should be chosen to give a positive value answer for the Area.

You can use determinants to solve a system of equations You can use determinants to solve a system of equations. The method is called Cramer’ rule and named after the Swiss mathematician Gabriel Cramer (1704-1752). The method uses the coefficients of the linear system in a clever way. ax + by = e In general the solution to the system is (x,y) cx + dy = f e b f d x= a b c d where a b c d = 0 and a e c f If we let A be the coefficient matrix of the linear system, notice this is just det A. y= a b c d

Use Cramer’s Rule to solve this system: 4x + 2y = 10 ax + by = e 5x + y = 17 1 cx + dy = f x= a b c d e f 10 2 17 1 (10)(1) –(17)(2) 10 - 34 = -24 -6 x= = = = 4 4 2 5 1 (4)(1) –(5)(2) 4 - 10 y= a b c d e f 4 10 5 17 (4)(17) –(5)(10) 68 - 50 = 18 -6 y= = = = -3 (4)(1) –(5)(2) 4 - 10 4 2 5 1 The system has a unique solution at (4,-3)

Solve the following system of equations using Cramer’s Rule: 6x + 4y = 10 ax + by = e 3x + 2y = 5 cx + dy = f x= a b c d e f 10 4 5 2 (10)(2) –(5)(4) 20 - 20 = x= = = (6)(2) –(3)(4) 12 - 12 6 4 3 2 Since, the determinant from the denominator is zero, and division by zero is not defined: THIS SYSTEM DOES NOT HAVE A UNIQUE SOLUTION and Cramer’s Rule can’t be used.

Cramer’ Rule can be use to solve a 3 x 3 system. Let A be the coefficient matrix of this linear system: If det A is not 0, then the system has exactly one solution. The solution is: