College Algebra Exam Review 180

# College Algebra Exam Review 180 - if one is transformed...

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190 3. PRODUCTS OF GROUPS by multiplication on the left by the m –by- m matrix E C ˇE i;j , where E is the m –by- m identity matrix, and E i;j is the matrix unit with a 1 in the .i;j/ position. E C ˇE i;j is invertible in Mat m . Z / with inverse E ± ˇE i;j . For example, for m D 4 , E C ˇE 2;4 D 2 6 6 4 1 0 0 0 0 1 0 ˇ 0 0 1 0 0 0 0 1 3 7 7 5 : The second type of elementary row operation interchanges two rows. The operation of interchanging the i –th and j –th rows is implemented by multiplication on the left by the m –by- m permutation matrix P i;j corre- sponding to the transposition .i;j/ . P i;j is its own inverse. For example, for m D 4 , P 2;4 D 2 6 6 4 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 3 7 7 5 : The third type of elementary row operation replaces some row a i with ± a i . This operation is it’s own inverse, and is implemented by left multi- plication by the diagonal matrix with 1 ’s on the diagonal except for a ± 1 in the .i;i/ position. Elementary column operations are analogous to elementary row oper- ations. They are implemented by right multiplication by invertible n –by– n matrices. We say that two matrices are row–equivalent
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Unformatted text preview: if one is transformed into the other by a sequence of elementary row operations; likewise, two matrices are column–equivalent if one is transformed into the other by a sequence of elementary column operations. Two matrices are equivalent if one is transformed into the other by a sequence of elementary row and column operations. In the following discussion, when we say that a is smaller than b , we mean that j a j ² j b j ; when we say that a is strictly smaller than b , we mean that j a j < j b j . Lemma 3.5.10. Suppose that A has nonzero entry ˛ in the .1;1/ position. (a) If there is a element ˇ in the ﬁrst row or column that is not di-visible by ˛ , then A is equivalent to a matrix with smaller .1;1/ entry. (b) If ˛ divides all entries in the ﬁrst row and column, then A is equivalent to a matrix with .1;1/ entry equal to ˛ and all other entries in the ﬁrst row and column equal to zero....
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