101st_tut1

101st_tut1 - (iii) If A is a square matrix such that A 2 ±...

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MATH1111/2010-11/ Tutorial I 1 MATH1111 Tutorial I 1. Consider the system 8 < : x 1 ± x 2 + x 3 ± x 4 + x 5 = 1 x 2 ± x 4 ± x 5 = ± 1 x 3 ± 2 x 4 + 3 x 5 = 2 : Determine with explanation whether the following is true. (a) There are two free variables. (b) The free variables must be x 4 and x 5 . 2. This question concerns how to prove or disprove a statement. We start with "common sense". For statments (a) and (b) below, think about what you need to do if you want to prove it. Also, think about how you can disprove it. (a) All apples are sweet. (b) Some apples are sweet. Prove or disprove the following. (i) All matrices X satisfying X 2 = 0 are zero matrices. (ii) Some square matrices B satisfying B 2 ± I = 0 are singular.
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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 &quot;false&quot; parts in Qn 3, ±gure out some conditions such that they become valid statements under the additional conditions....
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This note was uploaded on 05/04/2011 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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