Ch. 7 – Matrices and Systems of Equations

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Ch. 7 – Matrices and Systems of Equations

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Ch. 7 – Matrices and Systems of Equations

Augmented Matrix (dotted line optional) Matrices What is the order (dimensions) of each matrix? Augmented Matrix: a matrix derived from a system of equations 3 x 3 2 x 1 2 x 3 System Augmented Matrix (dotted line optional)

RREF it! Ex 1: This system is in REF. Put it in RREF. When a matrix has zeroes above and below each leading 1, it is in Reduced Row-Echelon Form (RREF) When a 3x4 matrix is in RREF, the last column will be the (x, y, z) solution to the system! Ex 1: This system is in REF. Put it in RREF. Instead of doing back-substitution, we can RREF the REF to get our solution! Add 2(E2) and E1 and replace for E1…

RREF it! Ex 1 (cont’d): Add -9(E3) and E1 and replace for E1…

RREF it! Ex 1 (cont’d): Check your answer on your calculator! Since our matrix is in RREF, our solution is… …( 1 , -1 , 2 ) Check your answer on your calculator!

RREF it! Ex 2: RREF this system to find the solution. Add -2(E1) and E2 and replace for E2… Add -(E1) and E3 and replace for E3…

RREF it! Ex 2 (cont’d): Add 5(E3) and E2 and replace for E2… Divide E2 by -11, then switch E2 and E3…

RREF it! Ex 2 (cont’d): Add -2(E2) and E1 and replace for E1…

RREF it! Ex 2 (cont’d): Answer: ( -3 , -1 , 1 ) Add 2(E3) and E2 and replace for E2… Answer: ( -3 , -1 , 1 )

RREF it! Ex 3: In the circuit diagram below, each junction (black dot) must have equal current flowing in and out. Set up a system of equations and RREF the system with your calculator to find the currents of each individual wire. Create an equation for each junction! 20 x2 x3 x1 x4 10 10 x5 Get variables on one side…

RREF it! Ex 3 (cont’d): Now put the system into a matrix and RREF it! We have infinite solutions! Let x5 = a, then x1 = a - 10, x2 = 30 - a, x3 = a - 10, and x4 = a + 10.