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4.5 Notes

# 4.5 Notes - numbers Problem 13 2 1 5-3 4 2 Using Cramer’s...

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4.5 Solving Systems of Equations by Determinants Goals: Evaluate determinants Use Cramer’s rule to solve a system of equations Purpose: Learn how computers solve a system of equations A square matrix has the same number of rows and columns. A determinant is a real number associated with a square matrix. The determinant of the square matrix 1 6 is 1 6 5 2 5 2 Determinant of a 2 x 2 Matrix: a b c d = ad – bc Problem 1: 3 5 -1 7

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Determinant of a 3 x 3 Matrix: Expand by the minors of a row or column + - + - + - + - + 1. Pick any row or column 2. Write these three elements with the proper sign, according to the array of signs above. If an element is in a position marked +, we add. If marked -, we subtract. 3. Multiply each element by its minor . To find the minor of an element, cross out the row and column it is in and find the determinant of the remaining

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Unformatted text preview: numbers. Problem 13: 2 1 5-3 4 2 Using Cramer’s Rule to solve a system of two equations: ax + by = h cx + dy = k D = a b = ad – bc c d D x = h b = hd – bk (x column replaced by constants) k d D y = a h = ak – hc (y column replaced by constants) c k Cramer’s Rule: x = x D D y = y D D as long as D is not 0 When D=0, there is no solution or infinite solutions so use another method! Problem 7: 2y – 4 = 0 x + 2y = 5 Cramer’s Rule for three equations: x D x D = y D y D = z D z D = as long as D is not 0 1 1 1 1 2 2 2 2 3 3 3 3 a x b y c z k a x b y c z k a x b y c z k + + = + + = + + = a b c k b c D = a b c D x = k b c a b c k b c a k c a b k D y = a k c D z = a b k a k c a b k Problem 19: 3x + z = -1-x – 3y + z = 7 3y + z = 5...
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4.5 Notes - numbers Problem 13 2 1 5-3 4 2 Using Cramer’s...

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