MATRICES AND VECTORS SPACE
- Space and subspace 7
- Dot Product, orthogonal projection - Cross product
6
- Crammer Method - Least square Method 5
- Determinant
- Cofactor expansion, Row Reduction Methods 4
- Homogenous system - Inverse matrix
3
- System of linear equations - Gauss-Jordan Elimination 2
- Matrix and Matrix’s Operations - Elementary Row Operations 1
- System of Differential equations 14
- Eigen Value, Eigen Vector - Diagonalization
13
- Kernel and Range of T 12
- Linear Transformation - Transformation matrix 11
- Inner Product Space : Norm, angle and distance - Orthogonal and orthonormal set, projection
- Gramm-Schimdt method 10
- Basis of Subspace
- Basis of Column space, Row space 9
- Linear independence - Linear Combination - Basis and Dimension 8
MATRICES
z
Matrix Notation
Definition
A matrix is a rectangular array of numbers. The numbers in the array are called entries in the matrix. The entry in row i and column j is denoted by the symbol aij
Size of matrix is described as in terms of the number of rows and columns it contains
A General m x n matrix is written as
mn m
m
n n
a a
a
a a
a
a a
a
K M M
M M
K K
2 1
2 22
21
1 12
MATRICES
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Matrix Notation
mn m
m
n n
a
a
a
a
a
a
a
a
a
2 1
2 22
21
1 12
11
Row 1
Column 2
MATRICES
=
4
2
3
1
A
z
Square Matrix of order n
a matrix with n rows and n columns
main diagonal
: a
11,a
22,…,a
nn
=
3
3
3
4
1
2
5
2
1
B
Order 2 Order 3
MATRICES
[
0
0
0
]
0
0
0
0
0
0
0
0
0
0
=
1
0
0
0
1
0
0
0
1
3 3x
I
z
Identity Matrices
A Square matrix with 1’s on the main diagonal and 0’s off the
main diagonal. Identity matrix is denoted by
I
z
Zero Matrices
A matrix all of whose entries are zero
=
1
0
0
1
2 2x
MATRICES
z
Triangular Matrices
A Square matrix in which all the entries above the main
diagonal are zero (
lower triangular
)
A Square matrix in which all the entries below the main
diagonal are zero (
upper triangular
)
MATRICES
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Reduced row-echelon form Matrices
Properties of Reduced row-echelon form
1. If a row does not consist entirely of zeros, then the first non-zero number in the row is 1. We call this (number 1) a leading 1
2. If there are any row that consist entirely of zeros (not-null row), then they grouped together at the bottom of the matrix
3. In any two successive not-null row, the leading 1 in the lower row occurs farther to the right than the leading 1in the higher row
MATRICES
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Reduced row-echelon form Matrices
Example of reduced row-echelon matrices
Example of matrices not in reduced row-echelon form
Properties 1
Operations on Matrices
Addition and Subtraction matrix
Scalar Multiples
Multiplying Matrices
Transpose of a Matrix
Trace of a Matrix
Operations on Matrices
Addition and Subtraction
Definition
If A and B are matrices of the same size, then the sum A+B is the matrix obtained by adding entries of B corresponding entries of A matrix and the difference A−B is the matrix obtained by subtracting
entries of B corresponding entries of A matrix . Example :
addition
Example :
Operations on Matrices
Definition
If A is any matrix and k is any scalar, then the product kA is the matrix obtained by multiplying each entry of the matrix A by k Example : scalar multiples
=
6
.
3
5
.
3
4
.
3
3
.
3
2
.
3
1
.
3
6
5
4
3
2
1
3
=
18
15
12
9
6
3
Operations on Matrices
Definition
If A is m x r matrix and B is r x n matrix, then the product A x B is the m x n whose entries are determined as follows. The entry in row i and column j of AB given by formula (AB)ij = ai1b1j+ ai2b2j+…+ airbrj
Example : multiplying matrices
Multiplying Matrices
Operations on Matrices
Definition
If A is any m x n matrix, then the transpose of A, denoted by AT, is defined to be the n x m matrix that the results from interchanging the
rows and columns of A
Example : transpose of a matrix
=
6
5
4
3
2
1
A
Transpose of a Matrix
=
6
3
5
2
4
1
T
Operations on Matrices
Definition
If A is n x n square matrix, then the trace of A, denoted by tr(A), is defined to be the sum of the entries on the main diagonal of A, or given by formula tr(A) = a11+a22+…+ann
Example : trace of a matrix
=
9
6
3
8
5
2
7
4
1
A
Trace of a Matrix
Operations on Matrices
Elementary Row Operations (ERO)
Elementary row operations are operations to eliminate matrix to be
reduced row-echelon form. When we have the (augmented matrix)
reduced row-echelon form, we will get solutions of system of linear equations easier. We will discuss this more at the next chapter There are three types of operations
1. Multiply a row through by a nonzero constant 2. Interchange two rows
Operations on Matrices
Steps in elimination
(Create reduced row-echelon form)We have matrix Amxn
• Go to first row, Change entry a11 to be 1 (choose the simplest operation)
• Change entries a21, a31,..,am1 to be 0
• Go to first next row, Change entry a22 to be 1 (we pass this step when a22=0 and go to the next entry a2k)
• Change entries a1k, a3k,..,amk to be 0
Operations on Matrices
Example elimination using ERO
A Reduced row echelon form ?
Leading 1 a11 not leading 1
not zero
Operations on Matrices
Example elimination using ERO (2)
~
Leading 1
Operations on Matrices
Example elimination using ERO (3)
Add 2 x row 3 to the second row
Leading 1
Reduced row echelon form
Notes
1. All ERO notations above is given to help students in
Operation in Matrices
a. A+B=B+A
b. A+(B+C)=(A+B)+C
c. A(BC)=(AB)C
d. A(B+C)=AB+AC
e. k(AB)=(kA)B ; k : skalar
f. (AT)T=A
g. (AB)T=BTAT
Exercises
Consider these matrices
1. Compute the following
a. BC b. A – BC c. CE
d. CB e. D – CB f. EET – A
2. Which matrices below are in reduced row echelon form ?
Exercises
3. Reduced matrices below to reduced row echelon form
