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
