Q.1 What is the order of a matrix with 3 rows and 4 columns?
3 × 3
4 × 3
3 × 4
4 × 4
Explanation - The order of a matrix is defined as number of rows × number of columns. Hence, 3 rows and 4 columns make a 3 × 4 matrix.
Correct answer is: 3 × 4
Q.2 Which of the following is a square matrix?
2 × 3
3 × 2
4 × 4
3 × 5
Explanation - A square matrix has the same number of rows and columns. Only 4 × 4 fits this definition.
Correct answer is: 4 × 4
Q.3 What is the transpose of a matrix?
Matrix obtained by swapping rows with columns
Matrix obtained by multiplying by itself
Matrix with all elements doubled
Matrix with rows reversed
Explanation - The transpose of a matrix is formed by interchanging rows and columns.
Correct answer is: Matrix obtained by swapping rows with columns
Q.4 If A is a 2 × 3 matrix and B is a 3 × 4 matrix, what is the order of AB?
2 × 4
3 × 3
4 × 2
3 × 2
Explanation - Matrix multiplication is possible when the number of columns of the first matrix equals the number of rows of the second. The result has dimensions (rows of first) × (columns of second).
Correct answer is: 2 × 4
Q.5 Which of the following is the identity matrix of order 2?
[[1, 1],[0, 1]]
[[1, 0],[0, 1]]
[[0, 1],[1, 0]]
[[2, 0],[0, 2]]
Explanation - The identity matrix has 1s on the main diagonal and 0s elsewhere.
Correct answer is: [[1, 0],[0, 1]]
Q.6 The determinant of a 2 × 2 matrix [[a, b],[c, d]] is:
a + d
ad - bc
ab + cd
a - d
Explanation - For a 2 × 2 matrix, determinant = ad - bc.
Correct answer is: ad - bc
Q.7 If a row of a determinant is multiplied by 2, the determinant is:
Doubled
Unchanged
Halved
Becomes zero
Explanation - Multiplying any row or column by a scalar multiplies the determinant by that scalar.
Correct answer is: Doubled
Q.8 Which type of matrix has all entries equal to zero?
Unit matrix
Zero matrix
Diagonal matrix
Identity matrix
Explanation - A zero matrix has all elements equal to zero.
Correct answer is: Zero matrix
Q.9 For which matrices is the determinant defined?
Rectangular matrices only
Square matrices only
Row matrices only
All types of matrices
Explanation - The determinant is defined only for square matrices.
Correct answer is: Square matrices only
Q.10 If the determinant of a matrix is zero, the matrix is:
Invertible
Singular
Orthogonal
Diagonal
Explanation - A square matrix with determinant zero is singular, meaning it has no inverse.
Correct answer is: Singular
Q.11 What is the inverse of the identity matrix I?
0
I
-I
Not defined
Explanation - The identity matrix is its own inverse.
Correct answer is: I
Q.12 The sum of a matrix and its transpose is always:
Skew-symmetric
Symmetric
Diagonal
Identity
Explanation - A matrix plus its transpose is always symmetric because entries across the diagonal become equal.
Correct answer is: Symmetric
Q.13 The product of a matrix and its inverse is:
Zero matrix
Identity matrix
Transpose matrix
Determinant
Explanation - By definition, A × A⁻¹ = I.
Correct answer is: Identity matrix
Q.14 If two rows of a determinant are identical, the determinant is:
Positive
Negative
Zero
One
Explanation - Determinant becomes zero if two rows or two columns are identical.
Correct answer is: Zero
Q.15 Which property is true for diagonal matrices?
All entries are zero
All diagonal entries are 1
All non-diagonal entries are zero
Rows and columns are identical
Explanation - A diagonal matrix has only diagonal elements possibly nonzero, others must be zero.
Correct answer is: All non-diagonal entries are zero
Q.16 If A is a 2 × 2 matrix, then det(Aᵀ) equals:
det(A)
2 det(A)
-det(A)
det(A)²
Explanation - The determinant of a transpose is the same as the determinant of the original matrix.
Correct answer is: det(A)
Q.17 What is the adjoint of a matrix?
Transpose of cofactor matrix
Inverse of matrix
Matrix with diagonal only
Determinant value
Explanation - The adjoint of a matrix is the transpose of its cofactor matrix.
Correct answer is: Transpose of cofactor matrix
Q.18 Which operation is not defined for two matrices of different orders?
Addition
Multiplication
Subtraction
None
Explanation - Addition and subtraction require matrices of the same order, while multiplication may still be defined.
Correct answer is: Addition
Q.19 If A is a diagonal matrix, then Aᵀ is:
Symmetric
Skew-symmetric
Zero matrix
Identity
Explanation - The transpose of a diagonal matrix is itself, hence symmetric.
Correct answer is: Symmetric
Q.20 What is the trace of a matrix?
Sum of all entries
Sum of diagonal elements
Product of diagonal elements
Determinant
Explanation - The trace of a square matrix is defined as the sum of its diagonal elements.
Correct answer is: Sum of diagonal elements
Q.21 If A is a 2 × 2 matrix with determinant 5, what is det(2A)?
10
20
25
40
Explanation - For a 2 × 2 matrix, multiplying all entries by k multiplies determinant by k². Hence det(2A) = 2² × 5 = 20.
Correct answer is: 20
Q.22 Which matrix represents a reflection about the x-axis?
[[1, 0],[0, 1]]
[[1, 0],[0, -1]]
[[0, 1],[1, 0]]
[[-1, 0],[0, 1]]
Explanation - Reflection in x-axis negates the y-coordinate, represented by [[1,0],[0,-1]].
Correct answer is: [[1, 0],[0, -1]]
Q.23 What is the rank of a zero matrix?
Equal to its order
0
1
Not defined
Explanation - The rank of a zero matrix is zero because it has no non-zero rows.
Correct answer is: 0
Q.24 If det(A) = 0, then system of linear equations AX = B is:
Always consistent
Always inconsistent
May be consistent or inconsistent
Has unique solution
Explanation - If det(A)=0, the system may have no solution or infinitely many solutions, but not a unique one.
Correct answer is: May be consistent or inconsistent
Q.25 Which of the following is not a property of determinants?
Interchanging two rows changes sign
If two rows are proportional, determinant is zero
Multiplying a row by scalar multiplies determinant by same scalar
det(AB) = det(A) + det(B)
Explanation - Correct property is det(AB) = det(A) × det(B).
Correct answer is: det(AB) = det(A) + det(B)