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Gravity
Terms in this set (81)
B34 means...
B matrix with dimensions 3x4
b34 means...
element in third row fourth column of B matrix
Zero Matrix
matrix where all elements are zeros
Zero matrix is the ____________ identity
additive
Matrix addition and subtraction can only be done if the matrices have the ______________, otherwise the operation is ____________
same dimensions; undefined/non-comformable
A+B is an m x n matrix PROPERTY
closure
A+B=0 PROPERTY
additive inverse
0A=0 PROPERTY
multiplicative property of the zero scalar and of the zero matrix
A24 x B47 is....
conformable, forms a 2x7 matrix
Is matrix multiplication always commutative?
No
Raising a matrix to a power is defined only for ___________ matrices
square
The transpose of a matrix A is created by writing the ________ as _________
rows; columns
Tranpose Matrix for:
123
456
789
147
258
369
Finite graph
a set of points (nodes) connected by a set of lines (edges)
Directed graph
a finite graph with direction associated with edges
Nodes & Edges
nodes-points; edges-lines
Adjacency Matrix
a way of representing a finite graph using a matrix
2 nodes connected by an edge are considered ___________
adjacent
Adjacency Matrix always have _________________
symmetry about main diagonal
___________ on main diagonal mean that _____________
Zeroes; no node is adjacent to itself
Having a 1 as an element in an adjacency matrix means...
1 way to get between those two points
Squaring the adjacency matrix for which points can be connected through 1 arrow results in...
adjacency matrix for which points can be connected through 2 arrows
A^-1 is the ______________________________ of A
multiplicative inverse
For a matrix to have an inverse it must be ____________
square
What is the determinant for 2x2 matrices?
ad-bc
Formula for Inverse of a 2x2 Matrix
If the determinant equals 0, it is...
undefined (determinant doesn't exist)
How can you find the multiplicative inverse of a matrix A (it exists) in square matrices greater than 2x2?
Augmented Matrix with identity
Row operations include
multiplying/dividing a row by nonzero constant, adding or subtracting a multiple or a row to another row, swapping rows
If a square matrix has an inverse it is called ____________ or ______________
invertible; non-singular
A matrix that does not have an inverse is called __________
singular
An inverse matrix is always ___________
unique
What rule could you use to find the determinant of a 3x3 matrix? Explain
Diagonal Rule: sum of product of elements of each down diagonal - sum of product of up diagonals
M subscript ab
delete ath and bth row of matrix and find determinant of that matrix
Cofactor of C subscript ab
Cab= (-1)^a+b x Cij
Minors and Cofactors Determinant Rule
Multiplying each cofactor of a row or column by its element and adding them together equals the determinant
Triangular Matrix
square matrix with all zero entries either above or below its main diagonal
Diagonal Matrix
a matrix that's both upper or lower triangular matrix
Upper Triangular Matrix
Equivalent systems have the ___________ solution set
same
Consistent Independent System
when two lines intersect at one point to produce exactly ONE solution
Consistent Dependent System
when two lines are coincident/identical it produces INFINITELY many solutions
Inconsistent System
when two lines are parallel they produce NO SOLUTION
Gaussian elimination
process of converting a system into row-echelon form using row operations
Row Echelon Form Rules (staircase form)
any rows consisting entirely of zeroes occur at bottom of matrix; for each row that doesn't consist entirely of zeros, the first nonzero entry is 1 (leading 1); for two successive nonzero rows, the leading 1 in the higher row is farther to the left
Reduced Row Echelon Form Rules
a reduced matrix where every column that has a leading 1 has zeros in every position above and below its leading 1
Gauss-Jordan elimination continues reduction process until ______________________ form is obtained
rref
Consistent and Independent MATRIX
reduced row matrix has the same number of non-zero rows as variables; left hand is USUALLY identity matrix; must have at least as many equations as variables
Consistent and Dependent MATRIX
reduced row matrix has more variables than non-zero rows; doesn't have to have row of zeros, but usually there is; can happen when you have less equations than variables
Inconsistent MATRIX
reduced row matrix has a row of zeros on the left but the right hand side of the row isn't 0
(x,y) reflected across X-axis
(x,-y)
(x,y) reflected across Y-axis
(-x,y)
To translate a 2x4 matrix 5 units right and 4 units down, use the following translation matrix
[5 5 5 5]
[-4 -4 -4 -4]
2x4 matrix means...
2 rows, 4 columns
Reflection over X-axis Matrix
[1 0]
[0 -1]
Reflection over Y-axis Matrix
[-1 0]
[0 1]
Reflection over y=x Matrix
[0 1]
[1 0]
90 Rotation Matrix
[0 -1]
[1 0]
180 Rotation Matrix
[-1 0]
[0 -1]
270 Rotation Matrix
[0 1]
[-1 0]
* Rotation Matrix
[cos
-sin
]
[sin
cos
]
Area of a Triangle Using Matrices
Determine if 3 Points are Collinear Using Matrices
|X1 Y1 1|
|X2 Y2 1| = 0
|X3 Y3 1|
Find the equation of a line passing through (x1,y1) and (x2,y2) Using Matrices
|x y 1|
|x1 y1 1| = 0
|x2 y2 1|
Cramer's Rule is uses determinants to _________________
solve a system of n linear equations in n unknowns
If...
a1x+b1y=d1
a2x+b2y=d2,
Cramer's Rule means x=..., y=...
X=Dx/D; y=Dy/D
With a 3x3 system, D=...
|a1 b1 c1|
|a2 b2 c2|
|a3 b3 c3|
With a 3x3 system, Dx=...
|d1 b1 c1|
|d2 b2 c2|
|d3 b3 c3|
In a system with Cramer's Rule, Dx is where...
answers (d) replaces the first column
In a system with Cramer's Rule, Dy is where...
answers (d) replaces the second column
In a system with Cramer's Rule, Dz is where...
answers (d) replaces the third column
The D matrix with Cramer's Rule is...
just the variable's coefficients (NOT THE ANSWERS)
If A, X, and C are matrices and AX=C, how do I solve for X?
Front multiply both sides with the inverse of matrix A
In a cryptogram, a 0 is equivalent to a...
space
To decode a cryptogram, the receiver of a message must first know the _________________
encoding matrix
Uncoded Matrix x Encoding Matrix A=
Coded Matrix
Coded Matrix x Decoding Matrix A^-1=
Decoded Matrix
Partial Fraction Decomposition can only be done if...
the degree of the numerator is less than the degree of the denominator
How to Graph a Point (x,y,z) in 3 Dimensions
Graph (x,y) and move it z units up/down
How to Graph the Equation of an equation with 3 variables like 12x+37+8z=24
Find the x, y, and z intercepts. Graph each point and connect them in a plane.
The three coordinate planes divided the space into ____________________
eight octants
;