1. Determinants
林育崧
蘇育劭

2. 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

4. 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

5. 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.

6. 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

7. 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) )

8. 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)

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

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

11. 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.

12. Cofactor Formula
Cofactor is useful when the matrix has many zeros

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

14. Formula for A-1 (using Cramer’s rule)

15. 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

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

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

18. 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.