C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes3-1

# C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes3-1 -...

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SECTION 3.1 DETERMINANTS A square matrix A has a determinant, denoted by det A . To calculate det A , we start with the 2 × 2 case: det " a b c d # = ad - bc For any matrix A , when we cross out row i and column j , we get a new matrix denoted by A ij . Then for any row i , det A = a i 1 ( - 1) i +1 det A i 1 + a i 2 ( - 1) i +2 det A i 2 + · · · + a in ( - 1) i + n det A in , and for any column j , det A = a 1 j ( - 1) 1+ j det A 1 j + a 2 j ( - 1) 2+ j det A 2 j + · · · + a nj ( - 1) n + j det A nj . Notice that the signs in the sums alternate, so we just have to get the ﬁrst one right! It is an incredible fact that no matter which row you choose, or which column you choose, you will get the very same result! The smaller determinants, together with the sign preceding them, are called cofactors, and the calculations are called

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Unformatted text preview: cofactor expansions of det A . Finally, a matrix is written with its entries enclosed by [ and ], and the determinant of a matrix is written with the entries of the matrix enclosed by | and | . EXAMPLE. Use two diﬀerent cofactor expansions to compute ± ± ± ± ± ± ± 1-2 1 3-4 5 2 0 ± ± ± ± ± ± ± . EXAMPLE. Explore the eﬀect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it aﬀects the determinant. " a b c d # , " a + kc b + kd c d # HOMEWORK: SECTION 3.1...
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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes3-1 -...

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