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Unformatted text preview: Math 313 Lecture #10 § 2.2: Properties of Determinants The Effect of Row Operations on Determinants. It would be great if we could first row reduce an n × n matrix A and then compute its determinant from the simpler row reduced matrix. But alas, this method is doomed to fail : an invertible matrix A is row equivalent to I and the determinant of I is 1, but is det( A ) = det( I )? What do the elementary row operations do to the determinant of a matrix? Case of R i ↔ R j . If A = a 11 a 12 a 21 a 22 and E = 0 1 1 0 , then det( EA ) = a 21 a 22 a 11 a 12 = a 12 a 21- a 11 a 22 =- det( A ) . Now let A = ( a ij ) be a 3 × 3 matrix and E the elementary matrix that switches row one with row two. Then (using cofactor expansion along the second row) det( EA ) = a 21 a 22 a 23 a 11 a 12 a 13 a 31 a 32 a 33 =- a 11 a 22 a 23 a 32 a 33 + a 12 a 21 a 23 a 31 a 33- a 13 a 21 a 22 a 31 a 32 =- det( A ) . Switching any two rows of A leads to the same result. What is the determinant of E ? It is det( E ) = det( EI ) =- det( I ) =- 1 . In general, if A is an n × n matrix and E is an elementary matrix corresponding to switching two rows, then det( EA ) =- det( A ) = det( E )det( A ). Case of αR i → R i . Let A be an n × n matrix and let E be the elementary matrix that corresponds to multiplying the i th through by α 6 = 0. Cofactor expansion along the i th row of EA gives det( EA ) = αa i 1 A i 1 + αa i 2 A i 2 + · · · + αa in A in = α ( a i 1 A i 1 + a i 2 A i 2 + · · · + a in A in ) = α det( A ) . In this calculation, notice that α does not appear in the cofactors A ij ; this is what makes this calculation work! What is the determinant of E ? It is det( E ) = det( EI ) = α det( I ) = α. Thus if E is an elementary matrix that corresponds to multiplying a row through by a nonzero α , then det( EA ) = α det( A ) = det( E )det( A )....
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