Facts for Exam #3
Math 2270, Spring 2005
Prove the following facts:
Chapter §6.
(1) (6.1.6) The determinant of an (upper or lower) triangular matrix is the product of its diagonal
entries.
(2) (6.2.1.a) Let A and B be two n × n matrices. If B is obtained from A by dividing a row of A
1
by a scalar k, then det(B) = det(A).
k
(3) (6.2.1.b) Let A and B be two n × n matrices. If B is obtained from A by a row swap, then
det(B) = − det(A).
(4) (6.2.1.c) Let A and B be two n × n matrices. If B is obtained from A by adding a multiple of
a row of A to another row, then det(B) = det(A).
(5) (6.2.2) A square matrix A is invertible if and only if det(A) 6= 0.
(6) (6.2.5) If A and B are similar matrices, then det(B) = det(A).
(7) (6.2.6) If A is an invertible matrix, then det(A−1 ) =
1
.
det(A)
(8) (6.2.7) If A is a square matrix, then det(AT ) = det(A).
(9) (6.3.3) Consider a 2 × 2 matrix A = [~v1 ~v2 ]. Then the area of the parallelogram defined by ~v1
and ~v2 is | det(A)|.
(10) (6.3.7) Consider the vectors ~v1 , ~v2 , . . . ,p
~vm in Rn . Then the m-volume of the m-parallelepiped
defined by the vectors ~v1 , ~v2 , . . . , ~vm is det(AT A), where A is the n × n matrix with columns
~v1 , ~v2 , . . . , ~vm .
In particular, if m = n this volume is | det(A)|.
(11) (6.3.10) Let A be an n × n invertible matrix. Then A−1 =
1
adj(A)
det(A)
Chapter §7.
(1) (7.1.2) The possible real eigenvalues of an orthogonal matrix are 1 and -1.
(2) (7.2.1) Consider an n × n matrix A and a scalar λ. Then λ is an eigenvalue of A if and only if
det(A − λIn ) = 0.
(3) (7.2.5) If A is an n × n matrix, then
det(A − λIn ) = (−λ)n + tr(A)(−λ)n−1 + · · · + det(A)
(4) (7.2.8) and (7.5.5) If an n × n matrix A has the eigenvalues λ1 , λ2 , . . . , λn (real or complex),
listed with their algebraic multiplicities, then
det(A) = λ1 λ2 . . . λn
and
tr(A) = λ1 + λ2 + · · · + λn
(5) (7.3.4.a) Consider an n × n matrix A. If we find a basis of each eigenspace of A and concatenate
all these bases, then the resulting eigenvectors ~v1 , . . . ~vs will be linear independent.
(6) (7.3.4.b) There exists an eigenbasis for an n × n matrix A if and only if the geometric multiplicities of the eigenvalues add up to n.