math 235 ch 2 practice

# math 235 ch 2 practice - (a If A is an invertible matrix...

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Math 235 - Section 6, Chapter 2 Practice Problems Adam Gamzon 1. If possible, compute the products AB and BA of the following matrices A and B . If it is not possible then state why this is so. (a) A = 1 2 3 0 - 2 1 7 0 5 and B = 1 3 0 - 1 0 4 7 2 - 3 1 1 2 (b) A = 4 - 2 3 1 2 1 1 - 5 0 and B = 1 3 0 9 3 1 - 3 8 - 1 2. If T ( x ) = Ax for an n × p matrix A and S ( x ) = Bx for a p × m matrix B , which of the compositions of functions is valid and why: T ( S ( x )) or S ( T ( x ))? (Assume n,p, and m are distinct integers.) For the composition that is valid, what is the corresponding matrix? 3. (a) Let A be a matrix for the reﬂection about the line y = x (in R 2 ). Use a geometric argument to explain why A 2 = I 2 . (b) Find the matrix A . 4. Determine if the following matrices are invertible. If they are, ﬁnd their inverse. (a) ± 4 - 2 2 1 ² (b) 3 - 2 3 0 2 1 1 - 5 4 (c) 1 4 - 2 7 1 2 2 - 1 2 - 3 0 1 5. True or false. Justify your answer in either case.

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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 , ( ABA-1 ) 3 = AB 3 A-1 . (c) If A = 4-2 3 1 2 1 1-5 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 ] = XY-Y X . Show that [ X, [ Y,Z ]] + [ Z, [ X,Y ]] + [ Y, [ Z,X ]] = 0 . 7. Let A be an n × n matrix such that A 4-5 A 3 + A-I 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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math 235 ch 2 practice - (a If A is an invertible matrix...

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