DrDelMath

SUMMARY
Chapter 7:  Matrices and Determinants

Section 7.1: Matrices and Systems of Equations

DEFINITION: A matrix is a rectangular array of numbers.
DEFINITION: The numbers in the array are called the entries of the matrix.
DEFINITION: If a matrix has m rows and n columns, the order of the matrix is .
Notation: In general we use capital letters to represent matricies and lower case letters to represent entries of a matrix. We use subscripts on the entries to indicate the row and column that the entry occupies.
Example:
                
When it is desirable to emphasize the notation used for individual entries we use instead of capital A to name the matrix.
Example:
                  

Terminology:
a)  If the number of rows n is the same as the number of columns, the matrix is called a square matrix and its order is n.
b) If a matrix has only one row, it is called a row matrix (or row vector).
c) If a matrix has only one column, it is called a column matrix (or column vector).
d) For a square matrix, the elements whose row index is the same as the column index () are called the main diagonal entries.
e) The matrix derived from a system of equations written in standard form with the constant term on the right is called the augmented matrix.
f ) The matrix consisting of the coefficients (in the same order) of a system of equations is called the coefficient matrix
Example:Consider the following system of equations:
                   
The corresponding coefficient matrix is:
                     
and the augmented matrix is:
                      

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: Operations with Matrices

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 i^throw and j^th column of the product will be the inner product of the i^th row from the first matrix and the j^th column of the second matrix.

Example: The entry in the 2^nd row and 3^rd column of the product will be the inner product of the 2^nd row and 3^rd column. Of course the 2^nd row comes from the first matrix and the 3^rd 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 I_n 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 matrix and I_n is the identity of order n, then I_nA = AI_n = A
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

Information about finding the inverse of a square matrix is found in the PDF file HERE.

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.

Section 7.5: Applications of Matrices and Determinants