Chapter 11 (Alg 2) Flashcards
Matrix
A rectangular arrangement of numbers in horizontal rows and vertical columns
Element
A number in the matrix
Dimensions Of A Matrix
“m x n”
Rows by columns
Equal (Matrices)
dimensions are the same and elements are in corresponding positions
Adding and Subtrating Matrices
only if they have the same dimensions
Scalar
In operations with matrices, a real number is called a scalar
Scalar Multiplication
Multiplying a Matrix by a scalar, you multiply each element by the scalar
Properties of Matrix Addition and Subraction
Let A, B and C be matrices with the same dimension, and let k be a scalar: APOA (A + B) + C = A + (B + C) CPOA A + B = B + A DPOA k(A + B) = kA + kB DPOS k(A - B) = kA - kB
Multiplying Matrices
If the # of columns in A is = the # rows in B; you can multiply them, it’s defined.
If the dimensions of A are m x n and dimensions of B are n x p, then the dimensions of the product AB are m x p
Maxtrix Multiplication
multiply the elements of each row of the first matrix by the elements of each column of the second matrix, and then add the products
Properties of Matrix Multiplication
Let A, B and C be matrices APOM A(BC) LDP A(B + C) = AB + AC RDP (A + B)C = AC + BC
Identity Matrix
A matrix that has 1’s on the main diagonal and 0’s elsewhere
{}
If A is any n x n matrix and I is the n x n identity matrix, then IA = AI = A
Inverse matrices
if the product of two matrices (in both orders) is the n x n identity matrix
{}
You can use the calc to find the inverse of a matrix
System of linear equations in standard form
AX = B
A is the coefficient matrix
X is the matrix of variables
B is the matrix of constants
Solution of the linear equations
X = A^-1 B
Multiply the inverse of matrix A by matrix B
Determinant
Associated with each sqaure (n x n) matrix is a real number called its determinant
Denoted by det A or |A|
The determinant of a 2 x 2 matrix
the difference of the products of the elements on the diagonals
Cramer’s Rule
using the determinant to solve a system of linear equations
Cramer Rule for a 2 x 2 system
Let A be the coefficient matrix of the linear system
{}
ax + by = e ; cx + dy = f {a b
c d}
- - - - - - - - - - - - -
If det A DNE 0, then the system has exactly one solution. The solution is:
|e b| |a e|
x = |f d| y = |c f|
——- ——-
|A| |A|
Sequence
A fucntion whose domain is a set of consectuive integers
Terms
Values in the range of the function
Finite, Infinite sequences
1,2,3,4 ; 1,2,3,4 . . .
Series
When the terms of a sequence are added together
Formulas for special series
n terms / 1 + 1 + 1 + 1 + 1 + 1 . . . = n
1 + 2 + 3 + 4 + 5 . . . = n(n + 1) over 2
1^2 + 2^2 + 3^2 + 4^2 . . . = n(n + 1)(2n + 1) over 6
Arithmetic Sequence
the difference between the terms is the same
Common difference
the constant difference is called the common difference and is denoted by d
Rule for an Arithmetic Sequence
The nth term of an arithmetic sequence with first term a sub 1 and common difference d can be found using the following rule.
a sub n = a sub 1 + (n - 1)d
Arithmetic series
adding the terms of an arithmetic sequence
The sum of a finite Arithmetic Series
The sum of the first n terms of an arithmetic series is given by the following formula
S sub n = n(a sub 1 + a sub n all over 2)
Geometric Sequence
the ratio of any term to the previous term is constant
Common Ratio
the constant ratio and is denoted by r
Rule for a Geometric Sequence
The nth term ofa geometric sequence with first term a sub 1 and common ratio r can be found using the following rule.
a sub n = a sub 1 r^(n-1)
Geometric Series
the expression formed by adding the terms of a geometric sequence
The sum of a finite Geometric Series
The sum of the first n terms of a geometric series with common ratio r when r DNE 1, is given by the following formula:
S sub n = a sub 1 (1 - r^n over 1 -r)
The Sum of an infinite Geometric Series
The sum of an infinite geometric series with first term a sub 1 and common ratio r is given by:
S = a sub 1 / 1 - r
|r| = 1, the series has no sum
Write a repeating Decimal as a Fraction
Example: 0.17171717…
0.17171717… = 0.17 + 0.0017 + 0.000017 . . .
= 17(0.01) + 17(0.01)^2 + 17(0.01)^3 . . .
= 17(0.01) over 1 - 0.01
= 17/99
