2130_Chapter7_Notes

When a1 exists this is same as multiplying a b by a1 a

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Unformatted text preview: · · ann xn bn If b = 0, then the system is homogeneous; otherwise, it is nonhomogeneous. 2 / 13 Nonsingular Case If the coefficient matrix A is nonsingular (i.e., det A = 0), then A is invertible, and we can solve Ax = b as follows: Ax = b =⇒ A−1 (Ax) = A−1 b =⇒ (A−1 A)x = A−1 b =⇒ Ix = A−1 b =⇒ x = A−1 b. The solution is therefore unique. Also, if b = 0, the unique solution to Ax = 0 is: x = A−1 0 = 0. Hence, if det A = 0, the only solution to Ax = 0 is the trivial solution x = 0. 3 / 13 2 How to Solve Even if A−1 exists, we need a method for solving Ax = b. Construct the augmented matrix: A | b . Then apply row operations, trying to convert A → I. When A−1 exists, this is same as multiplying A | b by A−1 : A | b −→ I | c = A−1 A | b = A−1 A | A−1 b = I | A− 1 b =⇒ c = A−1 b =⇒ x = c. If det A = 0, row operations on A | b will lead to one (or both) of the following rows: (a) 0 · · · 0 | 1 is equivalent to 0 = 1. (Two or more equations contradict each other.) T...
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