# notes3-1 - A . Finally, a matrix is written with its...

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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 cofactor expansions of det

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Unformatted text preview: 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 dierent cofactor expansions to compute 1-2 1 3-4 5 2 0 . EXAMPLE. Calculate 1 2-1 3 4 6 1 2 1 1 8 3-2 0 4 0 . EXAMPLE. Explore the eect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it aects the determinant. " a b c d # , " a + kc b + kd c d # HOMEWORK: SECTION 3.1...
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## This note was uploaded on 05/07/2010 for the course M 56945 taught by Professor Danielallcock during the Spring '09 term at University of Texas at Austin.

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notes3-1 - A . Finally, a matrix is written with its...

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