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北京邮电大学国际学院线性代数讲义Lectu.ppt

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Chapter Two Determinants Overview With each square matrix, it is possible to associate a real number called the determinant of the matrix and this value is very important in many kinds of applications. In this lecture, we will introduce the idea of determinant of matrices and show some properties of determinants. After that, we will show how the determinant of a matrix play an important role in some applications and give a theorem used to find the solution of a linear equations system. Lecture 5 The determinant of a matrix and their properties The Definition of Determinant Case I 1 ? 1 matrices If A ? ( a ) is a 1 ? 1 matrix, then A will have a multiplicative inverse if and only if a ? 0. Thus, if we define det( A) ? a then A will be nonsingular if and only if det( A) ? 0 . The Definition of Determinant Case II 2 ? 2 matrices ? a11 a12 ? Let A ? ? ? . We had known that A is nonsingular if and only if it is row a a 22 ? ? 21 equivalent to I . Then, if a11 ? 0 , we can test whether A is row equivalent to I by 1. Multiply the second row of A by a11 a12 ? ? a11 ?a a ? ? 11 21 a11a22 ? times the first row from the new second row 2. Subtract a21 a12 ? a11 ? ? 0 a a ?a a ? 11 22 21 12 ? ? Since a11 ? 0 , the resulting matrix will be row equivalent to I if and only if The Definition of Determinant Case II 2 ? 2 matrices a11a22 ? a21a12 ? 0 If a11 ? 0 , we can switch the two rows of A and the resulting matrix ? a21 a22 ? ? 0 a ? 12 ? ? is row equivalent to I if and only if a21a12 ? 0 . Thus, if we define det( A) ? a11a22 ? a12a21 for 2 ? 2 matrix, then A is nonsingular if and only if det( A) ? 0 . ? a11 We also denote the determinant of matrix ? ? a21 a12 ? a11 by a22 ? a21 ? a12 . a22 The Definition of Determinant Case III 3 ? 3 matrices ? a11 a12 a13 ? ? ? A 3 ? 3 matrix A ? ? a21 a22 a23 ? is nonsingular if and only if A is row ?a ? ? 31 a32 a33 ? equivalent to I . If we suppose that a11 ? 0 , ? a11 ? ? a21 ?a ? 31 a12 a22 a32 ? ? a a13 ? ? 11 ? ? a23 ? ? ? 0 ? a33 ? ? ? ? 0 ? a12 a11a22 ? a21a12 a11 a11a32 ? a31a12 a11 ? ? a13 ? a11a23 ? a21a13 ? ? a11 ? a11a33 ? a31a13 ? ? a11 ? The Definition of Determinant Case III 3 ? 3 matrices The resulting matrix will be row equivalent to I if and only if a11a22 ? a21a12 a11 a11a32 ? a31a12 a11 a11a23 ? a21a13 a11 a11a33 ? a31a13 a11 a11 ?0 This condition can be simplified to a11a22a33 ? a11a32a23 ? a12a21a33 ? a12a31a23 ? a13a21a32 ? a13a31a22 ? 0 Thus, if we define det( A) ? a11a22a33 ? a11



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