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Unformatted text preview: (a) If A is an invertible matrix then there are inﬁnitely many solutions to the system Ax = b . (b) For an invertible matrix A , ( ABA1 ) 3 = AB 3 A1 . (c) If A = 42 3 1 2 1 15 0 and B = 0 0 1 0 1 0 1 0 0 then A and B commute. I.e., AB = BA . 6. Deﬁne the following operation on n × n matrices: [ X,Y ] = XYY X . Show that [ X, [ Y,Z ]] + [ Z, [ X,Y ]] + [ Y, [ Z,X ]] = 0 . 7. Let A be an n × n matrix such that A 45 A 3 + AI n = 0. Show that A is invertible and express the inverse in terms of A . 1 8. Suppose that T : R 2 → R 2 is linear, that T ± 1 ² = ± 1 4 ² , and that T ± 1 1 ² = ± 2 5 ² . What is T ± 2 3 ² ? 2...
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 Spring '10
 STAFF
 Math, Linear Algebra, Algebra, Matrices, Invertible matrix, Adam Gamzon, Practice Problems Adam

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