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# hw4 - HOMEWORK 4 2 i 1 5i 3 3 2i 2i 1 Compute the...

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HOMEWORK 4 1. Compute the determinant of - 2 + i - 1 5 i 3 3 + 2 i - 2 i 4 i 0 1 + i . 2. We say that matrices A, B M n × n ( k ) are similar iff there exists an invertible C M n × n ( k ) such that A = CBC - 1 . Show that if A and B are similar matrices, then they have the same: (a) trace [Hint: You may use the results of problem 7 below without proof]; (b) determinant; (c) nullity; and (d) rank. where the nullity of a matrix A is the nullity of the corresponding transformation L A , and likewise for rank. 3. Let T, U : V W be isomorphisms, and α k \ { 0 } a non-zero scalar. (a) Is T + U : V W also an isomorphism? Prove or give a counterexample. (b) Is αT : V W also an isomorphism? Prove or give a counterexample. 4. (a) Suppose that T : V W and U : W X are invertible linear transformations. Show that UT : V X is also invertible, and that ( UT ) - 1 = T - 1 U - 1 . (b) Suppose that A, B M n × n ( k ) are invertible matrices. Show that AB is invertible, and that ( AB ) - 1 = B - 1 A - 1 .

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hw4 - HOMEWORK 4 2 i 1 5i 3 3 2i 2i 1 Compute the...

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