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Solutions – § 2.2 10. Find a 2 × 1 matrix x with entries not all zero such that A x = 4 x , where A = 4 1 0 2 . A x = 4 x 4 1 0 2 x 1 x 2 = 4 x 1 x 2 4 x 1 + x 2 2 x 2 = 4 x 1 4 x 2 4 x 1 + x 2 = 4 x 1 implies x 2 = 0. Moreover, if x 2 = 0, then x 1 can by anything. So one such vector is 1 0 . 14. In the following linear system, determine all values of a for which the resulting linear system has (a) no solution (b) a unique solution (c) infinitely many solutions 1 1 - 1 2 1 2 1 3 1 1 a 2 - 5 a R 1 - R 2 R 2 R 1 - R 3 R 3 1 1 - 1 2 0 - 1 - 2 - 1 0 0 4 - a 2 2 - a (a) We get no solution if we have the (3,3) entry equal zero and the (3,4) entry not equal to zero. Therefore a 2 - 4 = 0 and 2 - a = 0, thus a = - 2. (b) We get a unique solutions as long as neither the (3,3) entry nor the (3,4) entry are zero. Therefore a = ± 2. (c) We get infinitely many solutions if both the (3,3) and (3,4) entries are zero. Therefore
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