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Published byLaureen Edwards Modified over 8 years ago

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14.3 Matrix Equations and Matrix Solutions to 2x2 Systems OBJ: Use the Inverse of a 2 x 2 Matrix to solve a system of equations

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To find the inverse (A -1 ) of Matrix A EX: If A = 54 23 Find A -1 A 5 3 – 4 2 A -1 = 1 3-4 7 -2 5 Check your answer by finding A A -1 1 5 4 3-4 7 2 3 -2 5 If A = ab cd where A = ad – bc then A -1 = 1 d -b A -c a

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Use matrices to solve the given system of equations: 5x + 4y = -2 2x + 3y = -5 This system of two equations in two unknowns can be replaced by a single matrix equation. A X = C 5 4 x -2 2 3 y = -5

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Notice that the coefficient matrix: A = 5 4 2 3 is the matrix whose inverse was found in the previous example. A -1 = 1 3 -4 7 -2 5

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If you left-multiply both sides of the matrix equation by A -1, you get: 1 3 -4 5 4 x 1 3 -4 -2 7 -2 5 2 3 y = 7 -2 5 -5 3 -2 + -4 -5 -2 -2 + 5 -5 1 7 0 x 1 14 7 0 7 y = 7 -21 1 0 x = 2 0 1 y -3 x = 2 y -3

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Solve the matrix equation for X. 7 -2 3 -5 -1 5 -4 1 X – 0 4 = 2 -3 = + 7 -2 2 0 -4 1 X = 2 1 If you left-multiply both sides of the matrix equation by A -1 1 1 2 7 -2 1 1 2 2 0 -1 4 7 -4 1 X = -1 4 7 2 1 1 2 + 2 2 1 0 + 2 1 4 2 + 7 2 4 0 + 7 1 1 -1 0 1 6 2 -1 0 -1 X = -1 22 7 -6 -2 X = -22 -7

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