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101st_tut2

# 101st_tut2 - MATH1111/2010-11 Tutorial II 1 MATH1111...

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MATH1111/2010-11/ Tutorial II 1 MATH1111 Tutorial II 1. Let A and B be two matrices satisfying AB = BA . (a) Must A and B be square matrices of the same order? Explain. (b) Show that if the order is odd, then A or B is singular. (c) Must A or B be singular if the order is even? Justify your answer. 2. Let V be a vector space and x 2 V . A student says that it is ambiguous to write 2 x , because one may interpret 2 x in di erent ways, such as (2 x ) ; ( 1)(2 x ) ; ( 2) x ; 2( x ) ; (2 ( 1)) x ; What is your comment? Give your justi cation. 3. Are the following sets vector spaces with the indicated operations? If not, why not? (a) The set V of all polynomials of exact degree 3 together with 0; operations are the usual operations on polynomials. (b) The set V of 2 2 matrices whose determinant equal zero; operations are usual matrix operations. (c) The set V of 2 2 matrices whose column sums are equal; operations are usual matrix operations. (d) The set V = f 1 g consisiting of only the element 1 ( 2 R ) where both 1 + 1 and 1 (for any 2 R ) are de ned to be 1.

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101st_tut2 - MATH1111/2010-11 Tutorial II 1 MATH1111...

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