Linear Algebra MCQs – T4Tutorials.com

Question 1:

\[
\text{What is the determinant of the following matrix?}
\]
\[
A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
\]
\[
\text{(a) } 4, \quad \text{(b) } -2, \quad \text{(c) } 2, \quad \text{(d) } -1
\]
Answer: B

Question 2:

\[
\text{What is the rank of the matrix?}
\]
\[
A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 4 & 5 & 6 \end{pmatrix}
\]
\[
\text{(a) } 0, \quad \text{(b) } 1, \quad \text{(c) } 2, \quad \text{(d) } 3
\]
Answer: C

Question 3:

\[
\text{What is the inverse of the following matrix?}
\]
\[
A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}
\]
\[
\text{(a) } \begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix}, \quad \text{(b) } \begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix}
\]
Answer: A

Question 4:

\[
\text{Which of the following is a vector space?}
\]
\[
\text{(a) } \{ (x, y) : x + y = 1 \}, \quad \text{(b) } \{ (x, y) : x^2 + y^2 = 1 \}
\]
Answer: A

Question 5:

\[
\text{What is the characteristic polynomial of the matrix?}
\]
\[
A = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}
\]
\[
\text{(a) } \lambda^2 – 5\lambda + 6, \quad \text{(b) } \lambda^2 – 5\lambda + 2, \quad \text{(c) } \lambda^2 – 3\lambda + 2, \quad \text{(d) } \lambda^2 – 3\lambda + 3
\]
Answer: A

Question 6:

\[
\text{Find the eigenvalues of the matrix.}
\]
\[
A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}
\]
\[
\text{(a) } 2, 5, \quad \text{(b) } 1, 6, \quad \text{(c) } 3, 4, \quad \text{(d) } 2, 3
\]
Answer: A

Question 7:

\[
\text{Which of the following is a solution to the system of equations?}
\]
\[
x + y = 2, \quad 2x + 3y = 5
\]
\[
\text{(a) } x = 1, y = 1, \quad \text{(b) } x = 1, y = 2, \quad \text{(c) } x = 2, y = 1, \quad \text{(d) } x = 0, y = 2
\]
Answer: A

Question 8:

\[
\text{What is the null space of the matrix?}
\]
\[
A = \begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix}
\]
\[
\text{(a) } \{ (0, 0) \}, \quad \text{(b) } \{ (1, -2) \}, \quad \text{(c) } \{ (1, 2) \}, \quad \text{(d) } \{ (0, 1) \}
\]
Answer: B

Question 9:

\[
\text{Which of the following matrices is diagonalizable?}
\]
\[
A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}
\]
\[
\text{(a) } A, \quad \text{(b) } B, \quad \text{(c) both}, \quad \text{(d) neither}
\]
Answer: C

Question 10:

\[
\text{What is the solution to the system of equations?}
\]
\[
x + y = 3, \quad 2x – y = 1
\]
\[
\text{(a) } x = 1, y = 2, \quad \text{(b) } x = 2, y = 1, \quad \text{(c) } x = 3, y = 0, \quad \text{(d) } x = 0, y = 3
\]
Answer: B

Question 11:

\[
\text{What is the trace of the matrix?}
\]
\[
A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
\]
\[
\text{(a) } 4, \quad \text{(b) } 3, \quad \text{(c) } 5, \quad \text{(d) } 7
\]
Answer: C

Question 12:

\[
\text{What is the determinant of the matrix?}
\]
\[
A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}
\]
\[
\text{(a) } 0, \quad \text{(b) } 1, \quad \text{(c) } -1, \quad \text{(d) } 2
\]
Answer: A

Question 13:

\[
\text{Which of the following is a basis for the column space of the matrix?}
\]
\[
A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
\]
\[
\text{(a) } \{(1, 3), (2, 4)\}, \quad \text{(b) } \{(1, 2), (3, 4)\}, \quad \text{(c) } \{(1, 3)\}, \quad \text{(d) } \{(2, 4)\}
\]
Answer: A

Question 14:

\[
\text{What is the inverse of the matrix?}
\]
\[
A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
\]
\[
\text{(a) } \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}, \quad \text{(b) } \begin{pmatrix} -4 & 2 \\ 3 & -1 \end{pmatrix}
\]
Answer: A

Question 15:

\[
\text{What is the dimension of the solution space for the system?}
\]
\[
x + y + z = 0, \quad x – y + z = 0, \quad x + y – z = 0
\]
\[
\text{(a) } 1, \quad \text{(b) } 2, \quad \text{(c) } 3, \quad \text{(d) } 0
\]
Answer: A

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By: Prof. Dr. Fazal Rehman | Last updated: January 31, 2025