Week 2 matrices Flashcards
using systems of equations
1) use matrix equation
2) write out coefficient matrix, vectors of unknowns, constant vector
3) solve for X in the matrix equation, by multiplying by inverse A
4) Find inverse A
5) solve for X by subbing in calculated ANS
6) solve for x,y values
using systems of equations
1) notice equation 3 is double than equation 2, but = 0 not 18, so it’s inconsistent and no solution
2) another check, the determinant of A = 0
Question e
using systems of equations
1) to balance the equation, multiply each factor by different constants
2) write out an equation for each element (O, C, H) with their different constants
3) write it out as a system of equation (use matrix equation) - also choose which variable to be the constant
4) Use X= (A^-1)(b), find inverse A
5) sub ANS into X= (A^-1)(b) equation
ANS= inverse A, constant vector (b)
6) obtain 3 equations for the 3 unknown vectors (a,b,c)
7) choose a value for d so it gives a whole number for everything
8) sub back into original chemistry equation
Gaussian elimination goal?
To get augmented matrix into identity matrix, then you can read the b column (constant vector) to get the ANS
Question a
1) write the augmented matrix
2) use row reduction methods to get to identity matrix
3) match with x,y values with rows
What is the basic method for gaussian elimination
Question c
Question a, find inverse of A using gaussian elimination
0) Check if inverse A exists, if determinant = 0, then inverse doesn’t exist
1) write augmented matrix with identity matrix
2) perform gaussian elimination (Get LHS = Identity matrix, results in RHS= inverse)
3) Take out inverse A
4) take a factor out of the matrix so there’s only whole numbers in the matrix
what is row echelon?
where all elements below the main diagonal is 0
what is reduced row echelon form?
all remaining elements in a column containing 1 on the main diagonal is 0
Define rank
number of non-zero rows in the row echelon form of a matrix
where non-zero row = rows in a matrix which has at least 1 non-zero element in it
find the ranks and no. of free parameters
equation for the no. of free parameters
m - rank(coefficient matrix) = no. free parameters
m= no. of rows in the augmented matrix (includes the extra column)
what is the Rank properties
If:
what is a non-trivial solution?
X doesn’t equal 0
what is the eigenvalue problem?
what is the characteristic polynomial?
Find the eigenvalue and eigenvector
eigenvalue properties
What is the solution to this augmented matrix?
No solution, as system is inconsistent
(0 = -14)
Question c
1) use row reduction methods to achieve row echelon form
(Get all elements below the main diagonal to = 0)
2) note row 4, even has in its main diagonal position (column 4) = 0, this is fine
3) rank = 3
0) leave equation 4, find the 3 unknowns and then sub into equ 4 to get lambda
1) put int augmented matrix
2) row reduct to get coefficient matrix to = identity matrix
3) solve for x,y,z by comparing with RHS
4) sub ANS into equ 4 to get lambda
