Unformatted text preview: (iii) If A is a square matrix such that A 2 ± I = 0, then there exists a nonzero vector x such that Ax = x or Ax = ± x . 3. Prove or disprove the following. (a) ( A + B ) 2 = A 2 + 2 AB + B 2 for any two n ² n matrices. (b) If A and B are nonsingular n ² n matrices, then A + B is nonsingular. (c) Let A be a nonzero matrix. If AB = AC , then B = C . (d) Let A be invertible such that I + A is invertible. Then ( I + A ) ± 1 = I + A ± 1 . 4. To those "false" parts in Qn 3, ±gure out some conditions such that they become valid statements under the additional conditions....
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- Spring '10