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chapter3.page05

# chapter3.page05 - Ax = y with A = • 3-4-2 5 ‚ x = • x...

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1. VECTORS AND MATRICES 5 as Ax = y with A = a b c d , x = x 1 x 2 , and y = y 1 y 2 . Definition 1.4 . A square (ex. 2 × 2 or 3 × 3 ) matrix A is invertible if there is a matrix A - 1 such that AA - 1 = A - 1 A = I. Theorem 1.3 . Let A = a b c d be a 2 × 2 matrix, if A is invertible, we have A - 1 = 1 ad - bc d - b - c d So if A is invertible, to solve Ax = y , we need to simply multiply both sides with A - 1 , that is x = A - 1 y. Example 1.3 . Solve the system of equation 3 x 1 - 4 x 2 = 2 - 2 x 1 + 5 x 2 = 7 Solution The equation can be rewrite
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Unformatted text preview: Ax = y with A = • 3-4-2 5 ‚ , x = • x 1 x 2 ‚ , and y = • 2 7 ‚ . So in matrix form the system of equation is • 3-4-2 5 ‚• x 1 x 2 ‚ = • 2 7 ‚ . Now the inverse of A is A-1 = 1 3(5)-(-2)(-4) • 5 4 2 3 ‚ , so the solution is x = A-1 y = 1 7 • 5 4 2 3 ‚• 2 7 ‚ = • 38 7 25 7 ‚ a...
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