MAC 4105
Linear Algebra 1
4.0 Credits
Final Exam Review (Qns & Ans)
2025
©2025
, Question 1:
Let \( A \) be an \( n \times n \) matrix with \( n \) distinct
eigenvalues. Which of the following is always true?
A. \(A\) is diagonalizable.
B. \(A\) is symmetric.
C. \(A\) is invertible.
D. \(A\) is orthogonal.
Correct ANS: A. \(A\) is diagonalizable.
Rationale:
Distinct eigenvalues guarantee that the eigenspaces are one-
dimensional and that there exists a complete set of eigenvectors,
ensuring diagonalizability. Symmetry, invertibility, or orthogonality
are not ensured merely by having distinct eigenvalues.
---
Question 2:
Consider the linear transformation \(T : \mathbb{R}^2 \to
\mathbb{R}^2\) represented by
\[
A = \begin{bmatrix} 2 & 3 \\ 4 & 6 \end{bmatrix}.
\]
What is the rank of \(A\)?
A. 0
B. 1
C. 2
D. Not defined
Correct ANS: B. 1
Rationale:
The second row is a multiple of the first (in fact, \( [4 \; 6] = 2 \times
[2 \; 3] \)), so the rows are linearly dependent, and the rank (the
number of linearly independent rows) is 1.
---
Question 3:
©2025
, Which condition is necessary and sufficient for a square matrix \(A\)
to be invertible?
A. \(\det(A) = 0\)
B. \(A\) has a nontrivial null space
C. All eigenvalues of \(A\) are nonzero
D. \(A\) has at least one nonzero entry
Correct ANS: C. All eigenvalues of \(A\) are nonzero.
Rationale:
A square matrix is invertible if and only if its determinant is
nonzero—which is equivalent to having no zero eigenvalue.
---
Question 4:
Which factorization expresses a full‑column‑rank matrix \(A\) as
\[
A = QR,
\]
where \(Q\) is orthogonal and \(R\) is upper triangular?
A. LU factorization
B. QR factorization
C. Cholesky factorization
D. Singular Value Decomposition (SVD)
Correct ANS: B. QR factorization
Rationale:
The QR factorization decomposes a matrix into an orthogonal
matrix \(Q\) and an upper triangular matrix \(R\); it is especially
useful when \(A\) has full column rank.
---
Question 5:
For a real symmetric matrix \(A \in \mathbb{R}^{n\times n}\), which
theorem guarantees that \(A\) can be diagonalized by an
orthonormal basis?
©2025
Linear Algebra 1
4.0 Credits
Final Exam Review (Qns & Ans)
2025
©2025
, Question 1:
Let \( A \) be an \( n \times n \) matrix with \( n \) distinct
eigenvalues. Which of the following is always true?
A. \(A\) is diagonalizable.
B. \(A\) is symmetric.
C. \(A\) is invertible.
D. \(A\) is orthogonal.
Correct ANS: A. \(A\) is diagonalizable.
Rationale:
Distinct eigenvalues guarantee that the eigenspaces are one-
dimensional and that there exists a complete set of eigenvectors,
ensuring diagonalizability. Symmetry, invertibility, or orthogonality
are not ensured merely by having distinct eigenvalues.
---
Question 2:
Consider the linear transformation \(T : \mathbb{R}^2 \to
\mathbb{R}^2\) represented by
\[
A = \begin{bmatrix} 2 & 3 \\ 4 & 6 \end{bmatrix}.
\]
What is the rank of \(A\)?
A. 0
B. 1
C. 2
D. Not defined
Correct ANS: B. 1
Rationale:
The second row is a multiple of the first (in fact, \( [4 \; 6] = 2 \times
[2 \; 3] \)), so the rows are linearly dependent, and the rank (the
number of linearly independent rows) is 1.
---
Question 3:
©2025
, Which condition is necessary and sufficient for a square matrix \(A\)
to be invertible?
A. \(\det(A) = 0\)
B. \(A\) has a nontrivial null space
C. All eigenvalues of \(A\) are nonzero
D. \(A\) has at least one nonzero entry
Correct ANS: C. All eigenvalues of \(A\) are nonzero.
Rationale:
A square matrix is invertible if and only if its determinant is
nonzero—which is equivalent to having no zero eigenvalue.
---
Question 4:
Which factorization expresses a full‑column‑rank matrix \(A\) as
\[
A = QR,
\]
where \(Q\) is orthogonal and \(R\) is upper triangular?
A. LU factorization
B. QR factorization
C. Cholesky factorization
D. Singular Value Decomposition (SVD)
Correct ANS: B. QR factorization
Rationale:
The QR factorization decomposes a matrix into an orthogonal
matrix \(Q\) and an upper triangular matrix \(R\); it is especially
useful when \(A\) has full column rank.
---
Question 5:
For a real symmetric matrix \(A \in \mathbb{R}^{n\times n}\), which
theorem guarantees that \(A\) can be diagonalized by an
orthonormal basis?
©2025
