Section 1.7 Diagonal, Triangular, and
Symmetric Matrices
Topics:
I. Diagonal Matrices
II. Triangular Matrices
III.Symmetric Matrices
1
I. Diagonal Matrices
Definition : A square matrix in which all th
Department of Mathematics and Statistics
MATH 264.3 (01) 2010-2011 Term 1
Instructor:
Dr. Leslie Walter, 211 McLean Hall, ph: 966-6081, [email protected]
Lecture:
MWF 12:30 - 1:20 Thorv 110
Office
University of Saskatchewan
Department of Mathematics and Statistics
Math 264 (01)
October 18, 2010 Test #1 50 minutes
This is a format assessment. Cheating on on assessment is considered a serious o
University of Saskatchewan
Department of Mathematics and Statistics
Math 264 (01)
November 22, 2010 Test #2 50 minutes
This is a format assessment. Cheating on an assessment is considered a seri
Echelon systems and back substitution
Echelon systems
An echelon system is a system of linear equations with the
following properties:
1. Each equation is written so that the variables appear in the
s
Projections and orthogonal vectors
Theorem
For any non-zero vectors u and v in Rn , there exists unique vectors
v1 and v2 such that
v = v1 + v2
where v1 is parallel to u and v2 is orthogonal to u.
v
u
The n n identity matrix In
The standard basis vectors e1 , e2 , . . . , en
For i = 1, 2, . . . , n, the n 1 column matrix
0
.
.
ei =
0
1
0
.
.
i-th row
0
The entries of ei are all zero except th
University of Saskatchewan
Department of Mathematics and Statistics
Math 264 (01) Assignment #6
Due: December 3, 2010
Question 1. Is the vector w = (1, 5, 5) in the span of the vectors v1 = (1, 2, 1),
Chapter 2: Determinants
Determinants arise when one tries to solve
some linear systems
100 years ago they seemed more interesting
and more important than matrices
Determinants are now far from the
University of Saskatchewan
Department of Mathematics and Statistics
Math 264 (01) Assignment #1
Due: October 1, 2010
Question 1. Use the method of back-substitution to solve the system of linear equat
University of Saskatchewan
Department of Mathematics and Statistics
Math 264 (01) Assignment #2
Due: October 8, 2010
Question 1. For
10
A=
2 5
1 3
,
1 4
1 2
B=
.
compute A1 , B 1 , and (AB)1 .
Questio
University of Saskatchewan
Department of Mathematics and Statistics
Math 264 (01) Assignment #3
Due: October 15, 2010
Question 1.
consistent:
Under what conditions on b1 , b2 , b3 is the following sys
University of Saskatchewan
Department of Mathematics and Statistics
Math 264 (01) Assignment #5
Due: November 19, 2010
Question 1. Let u = (1, 1, 1) and v = (2, 1, 2).
6
(a) Find the components of 2u
University of Saskatchewan
Department of Mathematics and Statistics
Math 264 (01) Assignment #4
Due: November 5, 2010
Question 1. Using cofactor expansion, find
1 7
2 5
1 1
0 0
the determinant of th
Planes in 3-space
Normal vector to a plane
n
u
v
If x = a + su + tv is a vector equation for a plane then any
non-zero vector n that is orthogonal to both u and v is said to be
a normal vector to the
Solving multiple systems with the same
coefficients
Example
Find the solution to the system
2x3 =
1
2x1 + x2 + 2x3 =
3
x1
2x1 + x2 + 3x3 = 4
and the system
2x3 =
0
2x1 + x2 + 2x3 =
2
x1
2x1 + x2 + 3
More on invertible matrices: properties and
computing the inverse
Reduced echelon form of a square matrix
If A is square n n matrix and R is its reduced echelon form then
either:
(1) The rank of A is
Matrix operations and linear systems
Example
The system of linear equations
x2 +
2x1 + x2
x3 + 2x4 =
3
x3 + 2x4 = 1
3x1 x2 + 2x3 + 2x4 = 1
x1 +
x3 +
can be written as a matrix equation.
x4 =
1
0x1 +
Computing determinants: minor determinants
and cofactors
The expressions in the parentheses of the determinant of a 3 3
matrix
det(A) = a11 (a22 a33 a23 a32 ) a12 (a21 a33 a23 a31 ) + a13 (a21 a32 a22
Diagonal and triangular matrices
Diagonal matrices
If A is a square n n matrix, the entries A(i, i) are said to be on
the main diagonal of A.
A is called diagonal if all the entries of A above and bel
The determinant of a square matrix
Determinant of a 2 2 matrix
"
Recall that a 2 2 matrix A =
a b
c d
#
is invertible if and only if
ad bc 6= 0, and in this case,
A1
1
=
ad bc
"
d b
c
a
#
.
The determ
Matrix algebra
Matrices
A matrix is a rectangular array of numbers.
A=
a11
a21
.
.
a12
a22
.
.
a1n
a2n
.
.
am1 am2 amn
If a matrix has m rows and n columns, we say it has size m n.
The entry aij in th