MAS 3114
Computational Linear Algebra
4.0 Credits
Final Exam Review (Qns & Ans)
2025
©2025
, Question 1:
Let \( A \) be an \( n\times n \) matrix with distinct eigenvalues. In a
computational setting, why is diagonalization of \( A \)
advantageous?
A. It reduces computational complexity for matrix functions.
B. It guarantees numerical stability regardless of conditioning.
C. It always leads to a sparse representation.
D. It avoids the need for iterative methods.
Correct ANS: A. It reduces computational complexity for matrix
functions.
Rationale:
Diagonalization permits easy computation of matrix functions (e.g.,
exponentials, powers) by operating on the eigenvalues, thus
reducing overall complexity. However, it does not in itself guarantee
stability or sparsity.
---
Question 2:
In practical algorithms, the LU factorization of a matrix is often
combined with partial pivoting. What is the primary computational
benefit of using LU factorization with pivoting?
A. It increases the matrix’s sparsity.
B. It minimizes round-off errors and improves stability.
C. It guarantees that the matrix is diagonalizable.
D. It avoids the need for iterative solvers.
Correct ANS: B. It minimizes round-off errors and improves
stability.
Rationale:
Partial pivoting rearranges rows to keep pivot elements large,
thereby reducing round-off errors in floating point arithmetic and
enhancing the numerical stability of the algorithm.
---
©2025
, Question 3:
The condition number \(\kappa(A)\) of a matrix \(A\) (in the 2‑norm)
is defined as
\[
\kappa_2(A)=\|A\|_2\,\|A^{-1}\|_2.
\]
For which type of matrix is a high condition number most
problematic?
A. Diagonal matrices.
B. Well‑conditioned matrices.
C. Ill‑conditioned matrices.
D. Orthogonal matrices.
Correct ANS: C. Ill‑conditioned matrices.
Rationale:
A high condition number indicates that the matrix is ill‑conditioned,
meaning small perturbations in input can lead to large errors in the
solution, adversely affecting numerical algorithms.
---
Question 4:
Which iterative method is best suited for solving large, sparse
symmetric positive definite systems in computational linear
algebra?
A. Gaussian elimination.
B. Conjugate Gradient method.
C. Jacobi iteration.
D. LU factorization.
Correct ANS: B. Conjugate Gradient method.
Rationale:
The Conjugate Gradient (CG) method is specifically designed for
large, sparse, and symmetric positive definite matrices and is
favored in many applications for its efficiency and scalability.
---
©2025
, Question 5:
In computational routines, the Singular Value Decomposition (SVD)
is often used even though it is more expensive than other
factorizations. What is a primary advantage of the SVD?
A. It can handle non‑square matrices and provides optimal low‑rank
approximations.
B. It always produces a sparse factorization.
C. It automatically preconditions the matrix.
D. It requires no iterative methods.
Correct ANS: A. It can handle non‑square matrices and provides
optimal low‑rank approximations.
Rationale:
SVD is a powerful tool that not only decomposes any matrix
(square or rectangular) but also enables robust low‑rank
approximations, which are essential in areas such as data
compression and noise reduction.
---
Question 6:
Which factorization is particularly useful in computational settings
for solving least squares problems?
A. LU factorization
B. Cholesky factorization
C. QR factorization
D. Jordan canonical form
Correct ANS: C. QR factorization
Rationale:
QR factorization decomposes a matrix into an orthogonal (or
unitary) matrix and an upper triangular matrix, providing a
numerically stable method for solving least squares problems.
---
©2025
Computational Linear Algebra
4.0 Credits
Final Exam Review (Qns & Ans)
2025
©2025
, Question 1:
Let \( A \) be an \( n\times n \) matrix with distinct eigenvalues. In a
computational setting, why is diagonalization of \( A \)
advantageous?
A. It reduces computational complexity for matrix functions.
B. It guarantees numerical stability regardless of conditioning.
C. It always leads to a sparse representation.
D. It avoids the need for iterative methods.
Correct ANS: A. It reduces computational complexity for matrix
functions.
Rationale:
Diagonalization permits easy computation of matrix functions (e.g.,
exponentials, powers) by operating on the eigenvalues, thus
reducing overall complexity. However, it does not in itself guarantee
stability or sparsity.
---
Question 2:
In practical algorithms, the LU factorization of a matrix is often
combined with partial pivoting. What is the primary computational
benefit of using LU factorization with pivoting?
A. It increases the matrix’s sparsity.
B. It minimizes round-off errors and improves stability.
C. It guarantees that the matrix is diagonalizable.
D. It avoids the need for iterative solvers.
Correct ANS: B. It minimizes round-off errors and improves
stability.
Rationale:
Partial pivoting rearranges rows to keep pivot elements large,
thereby reducing round-off errors in floating point arithmetic and
enhancing the numerical stability of the algorithm.
---
©2025
, Question 3:
The condition number \(\kappa(A)\) of a matrix \(A\) (in the 2‑norm)
is defined as
\[
\kappa_2(A)=\|A\|_2\,\|A^{-1}\|_2.
\]
For which type of matrix is a high condition number most
problematic?
A. Diagonal matrices.
B. Well‑conditioned matrices.
C. Ill‑conditioned matrices.
D. Orthogonal matrices.
Correct ANS: C. Ill‑conditioned matrices.
Rationale:
A high condition number indicates that the matrix is ill‑conditioned,
meaning small perturbations in input can lead to large errors in the
solution, adversely affecting numerical algorithms.
---
Question 4:
Which iterative method is best suited for solving large, sparse
symmetric positive definite systems in computational linear
algebra?
A. Gaussian elimination.
B. Conjugate Gradient method.
C. Jacobi iteration.
D. LU factorization.
Correct ANS: B. Conjugate Gradient method.
Rationale:
The Conjugate Gradient (CG) method is specifically designed for
large, sparse, and symmetric positive definite matrices and is
favored in many applications for its efficiency and scalability.
---
©2025
, Question 5:
In computational routines, the Singular Value Decomposition (SVD)
is often used even though it is more expensive than other
factorizations. What is a primary advantage of the SVD?
A. It can handle non‑square matrices and provides optimal low‑rank
approximations.
B. It always produces a sparse factorization.
C. It automatically preconditions the matrix.
D. It requires no iterative methods.
Correct ANS: A. It can handle non‑square matrices and provides
optimal low‑rank approximations.
Rationale:
SVD is a powerful tool that not only decomposes any matrix
(square or rectangular) but also enables robust low‑rank
approximations, which are essential in areas such as data
compression and noise reduction.
---
Question 6:
Which factorization is particularly useful in computational settings
for solving least squares problems?
A. LU factorization
B. Cholesky factorization
C. QR factorization
D. Jordan canonical form
Correct ANS: C. QR factorization
Rationale:
QR factorization decomposes a matrix into an orthogonal (or
unitary) matrix and an upper triangular matrix, providing a
numerically stable method for solving least squares problems.
---
©2025
