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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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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