Exercises 28.1-11

Solution wanted!

Exercises 28.1-10

Solution wanted!

Exercises 28.1-9

Solution wanted!

Exercises 28.1-8

= = .

Let C = …

Exercises 28.1-7

Solution wanted!

Exercises 28.1-6

Exercises 28.1-5

These property can be proved directly by formula and property of permutation matrix.

Exercises 28.1-4

Product of two lower-triangular matrices is lower-triangular: Let A, B be two lower-triangular matrices and C = AB. = . For all i < j, , < 0. Hence the C is lower-triangular matrices.

Determinant: By Because matrix is lower-triangule, we can subtract row by the previous rows to make all columns other than diagonal become zero. By the property of determinant, the result diagonal matrix’s determinant is the same as original matrix.

Inverse of lower-trianguler matrix is lower-trianguler matrix: We can use adjugate matrix to prove this property

Exercise 28.1-3

Suppose that there are two matrix B and C such that AB = I = AC. We know that = 1 = and we can get n equations. We also can get the same equations from AC, because the coefficient of equation is the same, B and C must be the same.

Exercise 28.1-2

Let C = , D = ， = =