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Stitz-Zeager_College_Algebra_e-book

# We use the quadratic formula to solve 8u2 4u 2 0 and

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Unformatted text preview: trices.1 This means we will be able to evaluate the determinant of a 2 × 2 matrix as a sum of the determinants of 1 × 1 matrices; the determinant of a 3 × 3 matrix as a sum of the determinants of 2 × 2 matrices, and so forth. To explain how we will take an n × n matrix and distill from it an (n − 1) × (n − 1), we use the following notation. Definition 8.12. Given an n × n matrix A where n > 1, the matrix Aij is the (n − 1) × (n − 1) matrix formed by deleting the ith row of A and the j th column of A. For example, using the matrix A below, we ﬁnd the matrix A23 by deleting the second row and third column of A. 3 12 Delete R2 and C 3 A = 0 −1 5 − − − − − − − − − − − → A23 = 2 14 31 21 We are now in the position to deﬁne the determinant of a matrix. Definition 8.13. Given an n × n matrix A the determinant of A, denoted det(A), is deﬁned as follows • If n = 1, then A = [a11 ] and det(A) = det ([a11 ]) = a11 . • If n > 1, then A = [aij ]n×n and det(A) = det [aij ]n×n = a11 det (A11 ) − a12 det (A12 ) + − . . ....
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