Determinants 林育崧 蘇育劭

The Properties of the Determinant (1) The determinant of the n by n identity matrix is 1. (2) The determinant changes sign when two rows are exchanged (sign reversal)： (3) The determinant is a linear function of each row separately：   (3) Doesn’t mean det2I = 2 det I

The Properties of the Determinant (4) If two rows of A are equal , then det(A) = 0. (5) Subtracting a multiple of one row from another row leaves det(A) unchanged. (5) Conclusion 高斯消去從A到U 行列式不變

The Properties of the Determinant (6) A matrix with a row of zeros has det(A) = 0. By (4)(5) (7) If A is triangular then det A = product of diagonal entries.

The Properties of the Determinant (8) If A is singular then det A = 0.If A is invertible then det A != 0. If PA = LU then det P det A = det L det U (8) If A is singular elimination will produce a zero row

The Properties of the Determinant (9) |AB| =|A||B| (10) |A| = |AT| L,U,P has the same determinant asLT UT PT Important comment on columns : Every rule for rows can apply to the columns(Since(10) )

Three ways to compute determinants (1) Pivot Formula (Multiply the n pivots) (2) Big Formula (Add up n! terms) (3) Cofactor Formula (Combine n smaller determinants)

Pivot Formula PA = LU → det P det A = det L det U det A =

Big Formula There are possibly (3*2*1)=3! terms that may be non-zero.

Big Formula Matrices of n by n n! simple determinants that need to be summed up. Each simple determinant chooses one entry from every row and column The value of a simple determinant is the product times +1 or -1 The complete determinant of A is the sum of these n! simple determinants.

Cofactor Formula Cofactor is useful when the matrix has many zeros

Cramer’s Rule Solving Ax = b n x n system , need to evaluate (n + 1) determinants.

Formula for A-1 (using Cramer’s rule)

Direct proof of A-1 = cT /det A A A-1 = I → A cT = det (A) I How to explain the zores off the diagonal? *two equal rows

Applications of determiants Area is the absolute value of the determinant Area has the same properties as the determinant

(1) When , A = 1. (2) When rows are exchanged , the determinant reverses sign.(The absolute value stays the same) (3) Linearity

Cross Product Scalar Triple Product (b×c) ．a = = The volume of the parallelpiped If (b×c) ．a = 0 (1) a,b,c lie in the same plain (2) a,b,c are dependent. The matrix is singular. (3) zero volume.