Linear Algebra MCQs

\[
\textbf{MCQs on Linear Algebra with Answers}
\]

\[
\textbf{Q1: What is the determinant of the identity matrix of order \( n \)?}
\]
\[
\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } n \quad \text{(D) } \text{Undefined}
\]
\[
\textbf{Answer: (B) } 1
\]

\[
\textbf{Q2: A system of linear equations is consistent if:}
\]
\[
\text{(A) It has no solutions} \quad
\text{(B) It has exactly one solution} \quad
\text{(C) It has at least one solution} \quad
\text{(D) It has infinitely many solutions}
\]
\[
\textbf{Answer: (C) It has at least one solution}
\]

\[
\textbf{Q3: If \( A \) is a \( 3 \times 3 \) matrix, which of the following is true for \(\text{det}(A)\)?}
\]
\[
\text{(A) } \text{det}(A) = \text{det}(-A) \quad
\text{(B) } \text{det}(A) = -\text{det}(-A) \quad
\text{(C) } \text{det}(A) = 0 \quad
\text{(D) } \text{det}(A) > 0
\]
\[
\textbf{Answer: (A) } \text{det}(A) = \text{det}(-A)
\]

\[
\textbf{Q4: What is the rank of a \( 3 \times 4 \) zero matrix?}
\]
\[
\text{(A) } 0 \quad \text{(B) } 3 \quad \text{(C) } 4 \quad \text{(D) } \text{Undefined}
\]
\[
\textbf{Answer: (A) } 0
\]

\[
\textbf{Q5: For a square matrix \( A \), \( A^{-1} \) exists if and only if:}
\]
\[
\text{(A) } \text{det}(A) = 0 \quad
\text{(B) } \text{det}(A) \neq 0 \quad
\text{(C) } A \text{ is symmetric} \quad
\text{(D) } A \text{ is skew-symmetric}
\]
\[
\textbf{Answer: (B) } \text{det}(A) \neq 0
\]

\[
\textbf{Q6: In a vector space, the set of vectors is called linearly dependent if:}
\]
\[
\text{(A) One of the vectors can be written as a linear combination of the others} \quad
\text{(B) None of the vectors is a linear combination of the others} \quad
\text{(C) The dot product of any two vectors is zero} \quad
\text{(D) All vectors have equal magnitude}
\]
\[
\textbf{Answer: (A) One of the vectors can be written as a linear combination of the others}
\]

\[
\textbf{Q7: The eigenvalues of an identity matrix are:}
\]
\[
\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } \text{All equal to } n \quad \text{(D) } \text{All distinct}
\]
\[
\textbf{Answer: (B) } 1
\]

\[
\textbf{Q8: The null space of a matrix \( A \) is the set of all vectors \( x \) such that:}
\]
\[
\text{(A) } Ax = 0 \quad
\text{(B) } Ax = b \quad
\text{(C) } A^T x = 0 \quad
\text{(D) } A^T x = b
\]
\[
\textbf{Answer: (A) } Ax = 0
\]

\[
\textbf{Q9: What is the trace of a square matrix \( A \)?}
\]
\[
\text{(A) The product of its diagonal elements} \quad
\text{(B) The sum of its diagonal elements} \quad
\text{(C) The determinant of the matrix} \quad
\text{(D) The rank of the matrix}
\]
\[
\textbf{Answer: (B) The sum of its diagonal elements}
\]

\[
\textbf{Q10: Which of the following matrices is symmetric?}
\]
\[
\text{(A) } \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \quad
\text{(B) } \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \quad
\text{(C) } \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \quad
\text{(D) } \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
\]
\[
\textbf{Answer: (C) } \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}
\]