Homework 3 Solution - MAT 242 Written Homework#3 EP1.4...

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MAT 242 Written Homework #3 SOLUTIONS EP1.4, 1.5/H2.2, 2.3 Due: February 6/7 Solve the following problems, showing any necessary work. This includes row operations. 1. [1 point] Find all ordered pairs ( x, y ) such that 1 x 6 y · y - 5 - 10 x = y - 5 - 10 x · 1 x 6 y Solution: Multiplying the matrices together on each side of the equation yields: y - 10 x - 5 + x 2 6 y - 10 y - 30 + xy = y - 30 xy - 5 y - 10 + 6 x - 10 x + xy Converting this matrix equation into a system of equations produces: y - 10 x = y - 30 - 5 + x 2 = xy - 5 y - 4 y = - 10 + 6 x - 30 + xy = - 10 x + xy Equation 1 (or Equation 4) implies that x = 3 . Substituting this information into Equation 2 or 3 and solving for y produces y = - 2 . Thus, there is only one solution: (3 , - 2) . 2. Let A = 2 3 - 5 2 1 - 4 1 1 - 2 . a. [1 point] Use Gauss-Jordan Elimination to find A - 1 . Solution: 2 3 - 5 2 1 - 4 1 1 - 2 1 0 0 0 1 0 0 0 1 ------→ 1 - 3 1 2 - 3 2 1 - 4 1 1 - 2 1 0 - 1 0 1 0 0 0 1 --------→ 2 - 2 1 3 - 1 1 2 - 3 0 - 3 2 0 - 1 1 1 0 - 1 - 2 1 2 - 1 0 2 -------→ 2 - 4 3 1 2 - 3 0 1 - 2 0 - 1 1 1 0 - 1 2 1 - 6 - 1 0 2 ------→ 3 + 2 1 2 - 3 0 1 - 2 0 0 - 1 1 0 - 1 2 1 - 6 1 1 - 4 ----→ - 3 1 2 - 3 0 1 - 2 0 0 1 1 0 - 1 2 1 - 6 - 1 - 1 4 --------→ 1 + 3 3 2 + 2
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