of a Matrix
Suppose A =
Finding the Determinant of a 3 3 matrix
a11 a12 a13
Let A = a21 a22 a23
a31 a32 a33
Sarruss Rule for 3 3 matrices
Find the determinant of
A = 2 4 1
0 2 0
1.3 Matrix Arithmetic
Ordered pairs (x, y) = (1, 2)
Ordered triplets (x, y, z)
n ordered items, n-tuples
(x1 , x2 , , xn )
allows us to address a large range of problems.
What is the difference between a point and a vector?
3.3 Linear Independence
A homogeneous system such as
1 2 3
x2 = 0
can be viewed as a vector equation
9 = 0 .
x1 3 + x2 5 + x3
The vector equation has the trivial solution (x1 = 0, x2 = 0, x3 = 0),
but is this
1.6 Partitioned Matrices
A matrix can be partitioned into rows and columns. More generally, a matrix
can be partitioned into submatrices or blocks.
2 1 0
1 2 3
3 1 2
4 5 6
0 1 1
Suppose A is so large that you cannot access the ac
3.6 Row Space and Column Space
Two row equivalent matrices have the same row space.
Find a basis for the row space, a basis for the column space, and a basis for
1 0 1 1
A = 1 1 2 3
1 1 0 1
If A is a
Preliminaries: Sets and Complex Numbers
- Ricardo Appendix A.1
A set is a collection of objects, called elements.
to indicate that the collection is a set.
The set of natural numbers:
The set of all odd, positive integers
2.2 Properties of Determinants
Computing the determinant for some matrices is easy!
Otherwise, use cofactor expansion.
Finding Determinants using Row Operations
Let A =
, det(A) =
1. If B =
, det(B) =
2. If B =
3.1 Vector Spaces
Much of the framework we have developed for dealing with vectors in Rn can
be extended to treat other mathematical objects.
We can think of a vector space as a collection of objects that behave as
vectors do in Rn . The objects of such a
1.5 Elementary Matrices
Scaling - multiply a row by a nonzero constant
Interchange - switch two rows
Replacement - add multiple of one row to another row
An elementary matrix is one that is obtained by performin
2.3 Additional Topics and Applications
Let A be an n n matrix and Aij the cofactors, then
ai1 Aj1 + ai2 Aj2 + . + ain Ajn =
A11 A21 An1
A12 A22 An2
The adjoint of A is adjA = .
A1n A2n Ann
Section 1.2: Row Echelon Form
A matrix is in echelon form (or row echelon form) if it has the following
1. The first nonzero entry in each nonzero row is 1.
2. Each leading entry (i.e. left most nonzero entry) of a row is in a
Section 1.1: Systems of Linear Equations
A linear equation in n unknowns:
a1 x 1 + a2 x 2 + + an x n = b
where a1 , a2 , . . . an and b
x1 , x2 , .xn
Which equations are linear?
x2 = 2( 6 x1 ) + x3
4x1 5x2 + 2 = x1
4x1 6x2 = x1 x2
1.4 Matrix Algebra
Algebraic rules for real numbers may or may not extend to matrices.
Let A, B, and C be matrices, and let and be scalars.
(A + B) + C = A + (B +
3.4 Basis and Dimension
Span(v1 , v2 , , vn ) =
v1 , v2 , , vn are linearly independent if
c1 v1 + c2 v2 + + cn vn = 0
The vectors v1 , v2 , , vn form a basis for a vector space V if and only if
(i) v1 , v2 ,
Vector spaces formed from subsets of other vectors spaces are called subspaces.
A subspace of a vector space V is a subset S of V that has three properties:
a. The zero vector is in S.
b. For each x and y in S, x + y is in S.