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Unformatted text preview: AX = Y has no solution. 3. [12 points] Consider A = ± 2 3 3 4 ² . (a) [8 points] By any method, argue that A is invertible and compute A-1 . (b) [4 points] Write A as a product of elementary matrices. 4. [5 points] Find an example of 2 × 2 matrices A,B for which it is not true that ( A + B ) 2 = A 2 + 2 AB + B 2 . [Can you give a condition for A and B , so the last matrix formula would hold? You’ll get 5 extra points for the right answer.] 5. [5 points] Let A be an n × n matrix. Show that A is invertible if and only if A 2 is invertible. Bonus Problem [up to 10 points] Consider the equation X 2 =-I for 2 × 2 matrices. There are solutions with real coeﬃcient. Discover as many as you can. Can you ﬁnd inﬁnitely many? Give some more additional solutions with complex coeﬃcients. [You might also want to observe that if P is invertible 2 × 2, then if X solves, so does PXP-1 . Why?]...
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- Spring '10
- Linear Algebra, bonus problem, row-reduced echelon matrix