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(a) A is invertible.
(b) Ax = 0 has only the trivial solution.
(c) rref (A) = In .
(d) A = E1 E2 . . . Er for some elementary matrices E1 , E2 , . . . , Er .
(e) Ax = b has exactly one solution for every n × 1 column vector
(e) ⇒ (b): Take b = 0.
(b) ⇒ (a): Done.
(a) ⇒ (e): If A is invertible, then c = A−1 b and Ac = b. If c is
another solution, then c = A−1 Ac = A−1 b = c. Outline Number of Solutions Solving Multiple Systems More Results On the Inverse Example Example
Find the solution of the 1
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