Q.1 Which of the following is a vector space over the field of real numbers?
The set of all polynomials with real coefficients
The set of integers
The set of positive integers
The set of natural numbers
Explanation - A vector space must be closed under addition and scalar multiplication. Polynomials with real coefficients satisfy these conditions.
Correct answer is: The set of all polynomials with real coefficients
Q.2 If a matrix A has dimensions 3x4, what is the dimension of its transpose A^T?
4x3
3x4
4x4
3x3
Explanation - The transpose of a matrix swaps its rows and columns, so a 3x4 matrix becomes 4x3.
Correct answer is: 4x3
Q.3 The determinant of a triangular matrix is:
The sum of diagonal elements
The product of diagonal elements
Always zero
Cannot be determined
Explanation - For both upper and lower triangular matrices, the determinant equals the product of the diagonal entries.
Correct answer is: The product of diagonal elements
Q.4 If vectors u and v are linearly dependent, then:
u and v are perpendicular
One is a scalar multiple of the other
They form a basis of R^2
They are zero vectors
Explanation - Linear dependence implies that there exist scalars, not all zero, such that a linear combination of the vectors equals zero. For two vectors, this means one is a multiple of the other.
Correct answer is: One is a scalar multiple of the other
Q.5 Which of the following matrices is invertible?
A matrix with determinant 0
A square matrix with nonzero determinant
A rectangular matrix
A zero matrix
Explanation - A square matrix is invertible if and only if its determinant is non-zero.
Correct answer is: A square matrix with nonzero determinant
Q.6 The rank of a matrix is defined as:
The number of rows
The number of columns
The maximum number of linearly independent rows or columns
The determinant of the matrix
Explanation - The rank of a matrix is the dimension of the row space or column space, i.e., the number of linearly independent rows or columns.
Correct answer is: The maximum number of linearly independent rows or columns
Q.7 Which of the following is a property of a symmetric matrix?
A^T = -A
A^T = A
Determinant is always zero
All eigenvalues are zero
Explanation - A symmetric matrix is equal to its transpose, i.e., A^T = A.
Correct answer is: A^T = A
Q.8 Eigenvalues of a triangular matrix are:
Sum of diagonal elements
Product of diagonal elements
Diagonal elements
Always 1
Explanation - The eigenvalues of a triangular matrix (upper or lower) are exactly its diagonal elements.
Correct answer is: Diagonal elements
Q.9 Which of the following is true for an orthogonal matrix Q?
Q^T Q = I
Q^2 = 0
Det(Q) = 0
Q is always symmetric
Explanation - An orthogonal matrix satisfies Q^T Q = I, meaning its transpose is also its inverse.
Correct answer is: Q^T Q = I
Q.10 The solution to the homogeneous system Ax = 0 is:
Unique if A is invertible
Infinite if A is invertible
Never exists
Always infinite
Explanation - If A is invertible, the only solution is the trivial solution x = 0.
Correct answer is: Unique if A is invertible
Q.11 If two matrices A and B are similar, then:
They have the same determinant and eigenvalues
They are equal
Their product is identity
They are orthogonal
Explanation - Similar matrices represent the same linear transformation under different bases; they have the same eigenvalues and determinant.
Correct answer is: They have the same determinant and eigenvalues
Q.12 The dimension of the null space of a matrix A is called:
Rank
Nullity
Trace
Determinant
Explanation - The nullity of a matrix is the dimension of its null space (solution space of Ax=0).
Correct answer is: Nullity
Q.13 Which of the following operations is NOT defined for matrices?
Matrix addition
Matrix multiplication
Scalar multiplication
Matrix division
Explanation - Matrix division is not defined; instead, we multiply by the inverse when needed.
Correct answer is: Matrix division
Q.14 Which of the following matrices is idempotent?
A matrix where A^2 = A
A matrix where A^2 = 0
A matrix where A^T = A
A diagonal matrix
Explanation - By definition, an idempotent matrix satisfies A^2 = A.
Correct answer is: A matrix where A^2 = A
Q.15 The trace of a square matrix is:
The sum of all elements
The sum of diagonal elements
The product of diagonal elements
The determinant
Explanation - The trace of a square matrix is the sum of its diagonal entries.
Correct answer is: The sum of diagonal elements
Q.16 Two vectors in R^n are orthogonal if:
Their cross product is zero
Their dot product is zero
They are linearly dependent
They have the same magnitude
Explanation - Orthogonality of vectors is defined by the dot product being zero.
Correct answer is: Their dot product is zero
Q.17 Which of the following is a basis for R^3?
{(1,0,0), (0,1,0)}
{(1,0,0), (0,1,0), (0,0,1)}
{(1,2,3), (2,4,6)}
{(1,0,0), (0,1,0), (1,1,0), (0,0,1)}
Explanation - A basis must be linearly independent and span the space. Three independent vectors are needed to span R^3.
Correct answer is: {(1,0,0), (0,1,0), (0,0,1)}
Q.18 If a 2x2 matrix has eigenvalues 3 and 4, its determinant is:
7
12
1
0
Explanation - The determinant of a matrix equals the product of its eigenvalues: 3 * 4 = 12.
Correct answer is: 12
Q.19 If vectors v1, v2, ..., vn form an orthonormal set, then:
v_i · v_j = 0 for i≠j and |v_i|=1
v_i · v_j = 1 for all i,j
They are linearly dependent
Determinant of their matrix is 0
Explanation - Orthonormal vectors are orthogonal and each vector has unit length.
Correct answer is: v_i · v_j = 0 for i≠j and |v_i|=1
Q.20 The inverse of a 2x2 matrix [[a,b],[c,d]] is:
[[d,-b],[-c,a]]
(1/(ad-bc)) * [[d,-b],[-c,a]]
[[a,b],[c,d]]
Cannot be determined
Explanation - The formula for a 2x2 matrix inverse is (1/det(A)) * [[d,-b],[-c,a]], where det(A) = ad-bc.
Correct answer is: (1/(ad-bc)) * [[d,-b],[-c,a]]
Q.21 A matrix A is skew-symmetric if:
A^T = A
A^T = -A
Det(A)=0
All elements are zero
Explanation - By definition, a skew-symmetric matrix satisfies A^T = -A.
Correct answer is: A^T = -A
Q.22 If a 3x3 matrix has rank 2, the dimension of its null space is:
1
2
3
0
Explanation - By the Rank-Nullity Theorem: rank + nullity = number of columns. So, nullity = 3-2 = 1.
Correct answer is: 1
Q.23 Which of the following statements is true for a diagonalizable matrix?
It has n linearly independent eigenvectors
It is always symmetric
It has all zero eigenvalues
It is singular
Explanation - A matrix is diagonalizable if it has enough linearly independent eigenvectors to form a basis.
Correct answer is: It has n linearly independent eigenvectors
Q.24 Which of the following operations preserves linearity?
Matrix addition
Matrix inversion
Matrix determinant
Matrix transpose
Explanation - Matrix addition is linear because it satisfies additivity and scalar multiplication properties.
Correct answer is: Matrix addition
Q.25 The dot product of two vectors u = (1,2,3) and v = (4,5,6) is:
32
12
26
45
Explanation - Dot product u·v = 1*4 + 2*5 + 3*6 = 4+10+18 = 32.
Correct answer is: 32