Exam (elaborations)
Che 505 Final Exam Part 1 (Oral) With Complete Solutions Latest Update
- Course
- CHE 505
- Institution
- Arizona State University
Che 505 Final Exam Part 1 (Oral) With Complete Solutions Latest Update
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Solution 2024/2025Pepper
Che 505 Final Exam Part 1 (Oral) With
Complete Solutions Latest Update
Number of equations vs. number of unknowns
square systems/matrices have ANS✔✔ the same number of equations as
unknowns (m=n)
Cramers Rule ANS✔✔ tells if there is a unique and existing solution
If we want to solve Ax=b, find the kth element of the vector x by dividing
the determinant of A with the kth column replaced by vector b by the
determinant of A
When is a matrix singular, and what does singularity imply about the
solution ANS✔✔ det(A)=0
as long as a is not singular, a solution to Ax=b exists and is unique (1
solution)
Why is using the cramers rule impracticle ANS✔✔ scales as n factorial:
"combinatorial scaling"
slow alogorithims, avoid if possible
, Solution 2024/2025
Pepper
traingular matrices ANS✔✔ upper: every element below the main diagonal
is zero, nonzero elements on and above the main diagonal
lower: every element above the main diagonal is zero, nonzero elements on
and below the main diagonal
what is the determinant of a triangular matrix ANS✔✔ the product of all
terms on the main diagonal
how to solve Ax=b using a triangular matrix ANS✔✔ upper traingular matrix:
back solving
lower traingular matrix: forward solving
How can a matrix be transformed into a triangular matrix? ANS✔✔ Guassian
Elimination
Gaussian elimination ANS✔✔ main goal: get every value below the main
diagonal equal to zero using elemenentary row operation: multiply row by a
scalar and add rows together
generalized steps: every column from k=1 to n-1
every row from j=k+1 to n
multiply row k by -ajk/akk
add result to row j
augmented matrix ANS✔✔ add vector b as a new column in A
, Solution 2024/2025
Pepper
can do gaussian elimination of augmented matrix
Scaling of gaussian elimination ANS✔✔ consider the number of
multiplication and addition operations
O(n^3) polynomial scaling--fast algorithm
much better than combinatorial scaling (cramers rule)
How does guassian elimination affect the determinant of the matrix ANS✔✔
it doesnt: elementary row operations dont change the determinant
the determinant equals the determinant of the original matrix
pivoting ANS✔✔ using row swaps to ensure we can proceed with gaussian
elimination
swap with any row below where we currently are such that the new diagonal
term after swapping is nonzero
necessary when the diagonal term is exactly zero
useful when diagonal term is very small but nonzero (dividing by small
number can cause roundoff error)
swap row = partial pivoting
swap columns = full pivoting, the order of the x's change
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