Linear Algebra, 2020a

Section I. Definition 331
is the vector computed as this determinant.
⎛
⎞
⃗e 1 ⃗e 2 ⃗e 3
⃗x × ⃗y = det( ⎝x 1 x 2 x 3
⎠)
y 1 y 2 y 3
Note that the first row’s entries are vectors, the vectors from the standard basis for
R 3 . Show that the cross product of two vectors is perpendicular to each vector.
1.14 Prove that each statement holds for 2×2 matrices.
(a) The determinant of a product is the product of the determinants det(ST) =
det(S) · det(T).
(b) If T is invertible then the determinant of the inverse is the inverse of the
determinant det(T −1 )=(det(T)) −1 .
Matrices T and T ′ are similar if there is a nonsingular matrix P such that T ′ = PTP −1 .
(We shall look at this relationship in Chapter Five.) Show that similar 2×2 matrices
have the same determinant.
̌ 1.15 Prove that the area of this region in the plane
(
x2
y 2
)
(
x1
y 1
)
is equal to the value of this determinant.
( )
x1 x 2
det( )
y 1 y 2
Compare with this.
( )
x2 x 1
det( )
y 2 y 1
1.16 Prove that for 2×2 matrices, the determinant of a matrix equals the determinant
of its transpose. Does that also hold for 3×3 matrices?
1.17 Is the determinant function linear — is det(x · T + y · S) =x · det(T)+y · det(S)?
1.18 Show that if A is 3×3 then det(c · A) =c 3 · det(A) for any scalar c.
1.19 Which real numbers θ make (cos )
θ − sin θ
sin θ cos θ
singular? Explain geometrically.
? 1.20 [Am. Math. Mon., Apr. 1955] If a third order determinant has elements 1, 2,
..., 9, what is the maximum value it may have?
I.2 Properties of Determinants
We want a formula to determine whether an n×n matrix is nonsingular. We will
not begin by stating such a formula. Instead we will begin by considering, for