College Algebra Exam Review 182

# College Algebra Exam Review 182 - Q 2 Mat n Z such that PAQ...

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192 3. PRODUCTS OF GROUPS Again, the size of the .1;1/ entry cannot be reduced indeﬁnitely, so after some number of repetitions, we obtain a block diagonal matrix 2 6 6 6 4 d 1 0 ±±± 0 0 : : : 0 B 3 7 7 7 5 : whose .1;1/ entry d 1 divides all the other matrix entries. Step 3. By an appropriate inductive hypothesis, B is equivalent to a diagonal matrix diag .d 2 ;:::;d r ;0;:::;0/ , with d i dividing d j if 2 ² i ² j . The row and column operations effecting this equivalence do not change the ﬁrst row or ﬁrst column of the larger matrix, nor do they change the divisibility of all entries by d 1 . Thus A is equivalent to a diagonal matrix with the required divisibility properties. The nonzero diagonal entries can be made positive by row operations of type 3. n Example 3.5.11. (Greatest common divisor of several integers.) The di- agonalization procedure of Proposition 3.5.9 provides a means of comput- ing the greatest common divisor d of several nonzero integers a 1 ;:::;a n as well as integers t 1 ;:::;t n such that d D t 1 a 1 C ±±± t n a n . Let A de- note the row matrix A D .a 1 ;:::;a n / . By Propsition 3.5.9 , there exist an invertible matrix P 2 Mat 1 . Z / and an invertible matrix
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Unformatted text preview: Q 2 Mat n. Z / such that PAQ is a diagonal 1 –by– n matrix, PAQ D .d;0;:::;0/ , with d ³ . P is just multiplication by ˙ 1 , so we can absorb it into Q , giving AQ D .d;0;:::;0/ . Let .t 1 ;:::;t n / denote the entries of the ﬁrst column of Q . Then we have d D t 1 a 1 C ±±± t n a n , and d is in the subgroup of Z generated by a 1 ;:::;a n . On the other hand, let .b 1 ;:::;b n / denote the entries of the ﬁrst row of Q ± 1 . Then A D .d;0;:::;0/Q ± 1 imples that a i D db i for 1 ² i ² n . Therefore, d is nonzero, and is a common divisor of a 1 ;:::;a n . It follows that d is the greatest common divisor of a 1 ;:::;a n . Lemma 3.5.12. Let .v 1 ;:::;v n / be a sequence of n elements of Z n . Let P D Œv 1 ;:::;v n Ł be the n –by– n matrix whose j th column is v j . The following conditions are equivalent: (a) f v 1 ;:::;v n g is an basis of Z n . (b) f v 1 ;:::;v n g generates Z n . (c) P is invertible in Mat n . Z / ....
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