Linear Systems
The amount of solutions for consistent linear system of equations.
what is 1 or more?
The rules of row echelon form
1. All nonzero rows are above any rows of zeros
2. Each leading entry of a row is in a column to the right of the leading entries of any rows above it
3. Each leading entry of a row is equal to 1.
Scalar-valued function associated with a square matrix.
what is determinant?
find the 3 real numbers when
A+B=4
A+C=6
B+C=1
A=4.5
B=-0.5
C=1.5
Reduce this
[18 20 42]
[-3 -8 -21]
[-3 -4 -9]
[1 0 -1]
[0 1 3]
[0 0 0]
1. interchange the order in which the equations are listed
2. multiply any equation by a nonzero number
3. replace any equation with itself added to a multiple of another equation
The rules of REDUCED row echelon form
1. All nonzero rows are above any rows of zeros
2. Each leading entry of a row is in a column to the right of the leading entries of any rows above
3. Each leading entry of a row is equal to 1.
4. All entries in a column above and below a leading entry are 0.
LU decomposition equation
L=[1 0] [x 1]
U= [a b] [0 c]
A=[a b] [c d]
Augmented matrix of
5x-7y+3z=9
4x+10y+6z=-4
2x+6y-13z=-19
[5 -7 3 9]
[4 10 6 -4]
[2 6 -13 -19]
find the determinant
[-1 -3 4 -8]
[0 1 -3 3]
[0 0 -2 2]
[0 0 0 3]
A=6
This a system of equations.
what is a list of equations?
2x-4y+5z=-2
5x+4y-6z=-4
2x-4y-5z=-6
[2 -4 5 -2]
[5 4 -6 -4]
[2 -4 -5 -6]
A=mxn ?=nxm
what is A^T
compute:
[5 0 1] _ [2 0 -4]
[4 -3 -5] [-1 1 -5]
[-7 0 5]
[5 -4 0]
Calculate determinant
A=
[4 2 6]
[-2 -1 8]
[0 1 -3]
A=-44
this considered a system of homogenous equations.
what is when each equation is equal to 0?
rank is equal to the number of columns in the matrix.
what happens when the system has one unique solution?
(1/ad-bc)[d -b][-c a]
is this reduced row echelon form or just row echelon form?
[1 0 3]
[0 1 -5]
[0 0 1]
Row echelon form
Does A= [15 14] [-2 9] have an inverse? How do you know?
yes, the determinant is 163 and it is square.
solve:
6x+y=23
5x-6y=26
(x,y)=(4,-1)
inverse matrix rules.
1. square matrix
2. determinant not equal to 0
determinant of
A=[1 0][-1 -3]
-3
is the Reduced row echelon form or just row echelon form?
[1 0 0]
[0 3 0]
[0 0 1]
Row echelon form
find determinant for the system of equations.
6x+3y+7z=0
-x+y=0
18x+9y+18z=0
A=-27