1. Chapter 2 Determinants

2. 2.1 The Determinant Function The determine is a function that associates every square matrix with a number called the determinant of A, denoted by det(A). Definition: For a 2×2 matrix , if then det(A) = ad–bc • Properties: • The determinant function satisfies the following. • det(In) = 1 • If B is obtained from A by interchanging any two rows, then det(B) = –det(A). • If B is obtained from A by adding a multiple of one row to another, then det(B) = det(A). • If B is obtained from A by multiplying a number m to a row of A, then det(B) = mdet(A)

3. Theorem: If is a diagonal matrix, then det(D) = d11 d22 …dnn Theorem: If A is a triangular matrix (either upper or lower), then det(A) is also the product of its diagonal elements, det(A) = a11a22 …ann

4. 2.2 Properties of Determinants Theorem: If two rows of a square matrix A are the same, then det(A) = 0. Corollary: If all elements in one row of a square matrix A are zero then det(A) = 0. Theorem: A square matrix is invertible if and only if det(A) ≠ 0, and in this case det(A-1) = 1/det(A) Theorem: If A and B are both n×n matrices, then det(AB) = det(A)det(B) Warning: det(A + B) is NOT equal to det(A) + det(B) in general.

5. Theorem: If A is an n×n matrix and AT is its transpose, then det(AT) = det(A)

6. 2.3 Signed Elementary products Definition: A permutation of a set of integers is a reordering of those integers without omissions or repetitions. (another definition: A permutation of a set S of integers is a one-to-one onto function from S to itself. Example: There are 6 permutations of the set {1, 2, 3} namely, (1, 2, 3), (2, 1, 3), (3, 1, 2), (1, 3, 2), (2, 3, 1), (3, 2, 1) Remark: In general, if a set S has n elements, then there are n! many permutations of S.

7. Example: If A is a 4×4 matrix then a12 a23 a31 a44 is an elementary product. In general, an elementary product is of the form where (j1, j2, … , jn) is a permutation of the set {1, 2, … , n} Definition: An inversion occurs in a permutation (j1, j2, …, jn) whenever a larger number precedes a smaller one. A permutation is called even if it has an even number of inversions, and it is called odd if it has an odd number of inversions. Definition: An elementary product from a n×n matrix A is a product of n entries from A, exactly one from each row and one from each column.

8. Definition: A signed elementary product is an elementary product multiply by +1 or -1 depending on whether the permutation (j1, j2, … , jn) is even or odd. More precisely, we have (1) if (j1, j2, … , jn) is even if (j1, j2, … , jn) is odd. (2) Theorem: The any square matrix A, det(A) is the sum of all signed elementary products from A. Remark: For an n×n matrix, there are n! many elementary products, hence this is not an efficient way to compute the determinant of the matrix when n is large.

9. 2.4 Cofactor Expansion; Cramer’s Rule • Definition: • Given an n×n matrix A • The (i,j)-th minor of A, denoted by Mij , is the determinant of the (n - 1)×(n - 1) matrix obtained by deleting the i-th row and the j-th column of A. • The (i,j)-th cofactor of A, denoted by Cij, is (-1)i+j Mij Example:

10. Evaluating the Determinant by “Expansion by Cofactors” • Theorem: • Let A be an n×n matrix. Then det(A) can be computed by the following ways. • Expanding by cofactors along any row, i.e.det(A) = ai1Ci1 + ai2Ci2 + · · · +ainCin where 1 ≤ i ≤ n • Expanding by cofactors along any column, i.e.det(A) = a1jC1j + a2jC2j+ · · · +anjCnjwhere 1 ≤ j ≤ n

11. The Adjoint and a Theoretical Formula for A–1 Definition: If A is an n×n matrix, the adjoint of A, denoted by adj(A), is the transpose of the matrix of cofactors, Theorem: If A is an n×n matrix, then A× adj(A) = (detA) In Thus if det(A) ≠ 0, A–1 exists and

12. Cramer’s Rule Let AX = B be a system of n linear equations in n unknowns, where det(A) ≠ 0. The unique solution to this system is given by where etc.