DrDelMath
College Algebra by Stewart/Redlin/Watson
SUMMARY
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DrDelMath College Algebra by Stewart/Redlin/Watson |
Chapter 7: Matrices and Determinants
Section 7.1: Matrices and Systems of Equations
Some additional links
1. Summary of elementary matrix theory
2. More about matrices
3. Elementary Row Operations
4. Elementary Row Operations Calculator
Example:
When it is desirable to emphasize the notation used for individual entries we use [aij]
instead of capital A to name the matrix.
Example:
Example:
Consider the following system of equations:
The corresponding coefficient matrix is:
and the augmented matrix is:
Example:
Elementary
Row Operations:
1. Interchange two rows
2. Multiply a row by a non-zero constant and replace that row with the product.
3. Add a multiple of a row to another row and replace one but not both of the rows with that sum.
DEFINITION:
Two matrices are row equivalent
if one can be obtained from the other by a sequence of elementary row operations.
Section 7.2: The Algebra of Matrices
HERE ARE SOME ADDITIONAL LINKS
3. Matrix
Multiplication
4. Matrix
multiplication calculator
DEFINITION:
Two matrices
and 
are
equal if they have the same order
and their corresponding entries are equal.
A
more
formal definition is:
DEFINITION:
Two matrices 
and
are equal if and only if
1. m = p,
2. n = q, and
3. 
DEFINITION:
The sum
of two matrices A =
and B =
of the same order 
is
the matrix
defined by A + B = 
Example:
Note:
The
sum of two matrices with different orders is not defined.
DEFINITION:
A scalar
is a real number.
DEFINITION:
If A = is
a matrix and c is a scalar then the scalar
multiple is defined by cA = 
DEFINITION:
The
opposite of a matrix
A = is
the scalar multiple (-1)A = 
DEFINITION:
The
matrix
whose entries are all 0 is called a zero
matrix and it is the additive
identity for the set of matricies.
The zero matrix is frequently denoted by O.
Properties of Matrix Addition and Scalar
Multiplication:
If A, B, and C are
and both c and d are scalars, then
1. Matrix addition is commutative
A + B = B + A
2. Matrix addition is associative
A + (B + C) = (A + B) + C
3. Scalar Multiplication is associative
(cd)A = c(dA)
4. The number 1 is the Scalar Identity
1A = A
5. Scalar multiplication distributes over matrix addition
c(A + B) = cA + cB
6. Scalar addition distributes over scalar multiplication
(c + d)A = cA + dA
7. The zero matrix O is the additive identity
A + O = O + A = A
DEFINITION:
If X = 
is a row matrix with n entries and Y = 
is a column matrix with n entries, then the inner product (sometimes called dot product) 
is a scalar computed according to the rule:
= 
Note: A easy way of remembering this rule is to observe that this is just the sum of the products of corresponding entries.
DEFINITION:
If A =
is an matrix
and B = 
is an 
matrix, then the product AB is defined as the 
matrix AB = 
.
Note: This rule tells us that the entry in the ithrow and jth column of the product will be the inner product of the ith row from the first matrix and the jth column of the second matrix.
Example: The entry in the 2nd row and 3rd column of the product will be the inner product of the 2nd row and 3rd column. Of course the 2nd row comes from the first matrix and the 3rd column comes from the second matrix.
DEFINITION:
The identity matrix of
order n is the 
matrix whose main diagonal entries are 1 and all other entries are 0. The identity
matrix of order n is usually denoted by In or simply I when the order
is obvious.
Properties
of Matrix Multiplication:
If A, B, and C arematricies and c is a scalar, then
1. Matrix multiplication is associative
A(BC) = (AB)C
2. Matrix multiplication on the left distributes over matrix additon
A(B + C) = AB + AC
3. Matrix multiplication on the right distributes over matrix additon
(A + B)C = AC + BC
4. Multiplicative Identity
If A is an
5. Scalar multiplication is associative with respect to matrix multiplication
c(A + B) = cA + cB
Important Note: It is important to remember that matrix multiplication is not commutative. In general AB is not equal to BA even if both products are defined.
Section 7.3: The Inverse of a Square Matrix and Matrix Equations
Information about finding the inverse of a square matrix is found in the PDF file HERE.
Here are some links to addeitional websites about Matrix INVERSION
1. Basic
Lesson for Matrix Inversion
2. Finding
the Inverse of a Matrix
3. Finding
the inverse of a Matrix
Section 7.4: The Determinant of a Square Matrix
DEFINITION: The determinant is a function whose domain is the set of square matricies and whose range is the Real numbers. The range value associated with a particular matrix is called the determinant of that matrix.
Comment: The name of the determinant function is det. Vertical bars (such as used in absolute value) are also used to denote the determinant function. For example if A is a matrix whose determinant is 5, we would write det(A) = 5 or | A | = 5.
