Linear Algebra

Section I. Definition 2931.18 Which real numbers θ make (cosθ)− sin θsin θ cos θsingular? Explain geometrically.? 1.19 If a third order determinant has elements 1, 2, . . . , 9, what is the maximumvalue it may have? [Am. Math. Mon., Apr. 1955]I.2 Properties of DeterminantsAs described above, we want a formula to determine whether an n×n matrixis nonsingular. We will not begin by stating such a formula. Instead, we willbegin by considering the function that such a formula calculates. We will definethe function by its properties, then prove that the function with these propertiesexist and is unique and also describe formulas that compute this function.(Because we will show that the function exists and is unique, from the start wewill say ‘det(T )’ instead of ‘if there is a determinant function then det(T )’ and‘the determinant’ instead of ‘any determinant’.)2.1 Definition A n×n determinant is a function det: M n×n → R such that(1) det(⃗ρ 1 , . . . , k · ⃗ρ i + ⃗ρ j , . . . , ⃗ρ n ) = det(⃗ρ 1 , . . . , ⃗ρ j , . . . , ⃗ρ n ) for i ≠ j(2) det(⃗ρ 1 , . . . , ⃗ρ j , . . . , ⃗ρ i , . . . , ⃗ρ n ) = − det(⃗ρ 1 , . . . , ⃗ρ i , . . . , ⃗ρ j , . . . , ⃗ρ n ) for i ≠ j(3) det(⃗ρ 1 , . . . , k⃗ρ i , . . . , ⃗ρ n ) = k · det(⃗ρ 1 , . . . , ⃗ρ i , . . . , ⃗ρ n ) for k ≠ 0(4) det(I) = 1 where I is an identity matrix(the ⃗ρ ’s are the rows of the matrix). We often write |T | for det(T ).2.2 Remark Property (2) is redundant sinceT ρi+ρj−→ −ρj+ρi−→ρi+ρj−→ −→ −ρi ˆTswaps rows i and j. It is listed only for convenience.The first result shows that a function satisfying these conditions gives acriteria for nonsingularity. (Its last sentence is that, in the context of the firstthree conditions, (4) is equivalent to the condition that the determinant of anechelon form matrix is the product down the diagonal.)2.3 Lemma A matrix with two identical rows has a determinant of zero. Amatrix with a zero row has a determinant of zero. A matrix is nonsingular ifand only if its determinant is nonzero. The determinant of an echelon formmatrix is the product down its diagonal.