Q.1 Which of the following is a square matrix?
2x3 matrix
3x2 matrix
3x3 matrix
4x2 matrix
Explanation - A square matrix has the same number of rows and columns. Only 3x3 fits this definition.
Correct answer is: 3x3 matrix
Q.2 If A is a 2x3 matrix and B is a 3x2 matrix, what will be the order of AB?
2x2
3x3
2x3
3x2
Explanation - Matrix multiplication results in the order (rows of first) × (columns of second). Hence 2x2.
Correct answer is: 2x2
Q.3 Which element is located at position a23 in a matrix?
Row 2, Column 3
Row 3, Column 2
Row 1, Column 2
Row 2, Column 1
Explanation - The notation aᵢⱼ denotes element in ith row and jth column.
Correct answer is: Row 2, Column 3
Q.4 What is the determinant of the matrix [[1,2],[3,4]]?
-2
-5
2
5
Explanation - Determinant of [[a,b],[c,d]] = ad - bc. Here, (1*4 - 2*3) = -2.
Correct answer is: -2
Q.5 Which property is true for determinants?
Det(A) = Det(Aᵀ)
Det(A) = -Det(Aᵀ)
Det(A) = 0 always
Det(A) changes randomly
Explanation - The determinant of a matrix equals the determinant of its transpose.
Correct answer is: Det(A) = Det(Aᵀ)
Q.6 What is the adjoint of a square matrix?
Transpose of cofactor matrix
Inverse of the matrix
Determinant of the matrix
Product of rows
Explanation - Adjoint is defined as the transpose of the cofactor matrix.
Correct answer is: Transpose of cofactor matrix
Q.7 For a singular matrix A, which is true?
Det(A) = 0
Det(A) ≠ 0
Inverse exists
Rank = order
Explanation - A singular matrix has zero determinant, hence inverse does not exist.
Correct answer is: Det(A) = 0
Q.8 If a row of a determinant is multiplied by 0, what is the determinant?
Unchanged
Zero
Negative
Doubled
Explanation - If any row or column is zero, the determinant becomes zero.
Correct answer is: Zero
Q.9 Which matrix has only diagonal elements non-zero?
Row matrix
Diagonal matrix
Null matrix
Identity matrix
Explanation - A diagonal matrix has non-zero elements only on its main diagonal.
Correct answer is: Diagonal matrix
Q.10 The determinant of an identity matrix of order n is:
0
n
1
n!
Explanation - The determinant of an identity matrix is always 1, irrespective of its order.
Correct answer is: 1
Q.11 If det(A) = 5, what is det(2A) for a 2x2 matrix?
10
20
25
40
Explanation - For an n×n matrix, det(kA) = kⁿ det(A). Here n=2, so (2² * 5) = 20.
Correct answer is: 20
Q.12 Which matrix has only one row?
Column matrix
Row matrix
Square matrix
Diagonal matrix
Explanation - A matrix with only one row is called a row matrix.
Correct answer is: Row matrix
Q.13 If A = [[2,0],[0,3]], then det(A) = ?
6
5
0
2
Explanation - For diagonal matrices, determinant is the product of diagonal elements: 2*3 = 6.
Correct answer is: 6
Q.14 What is the rank of a zero matrix?
0
1
n
Undefined
Explanation - The zero matrix has no non-zero rows or columns, so its rank is 0.
Correct answer is: 0
Q.15 If det(A) = 0, then A⁻¹ exists?
Yes
No
Sometimes
Only if A is symmetric
Explanation - A matrix has an inverse only if its determinant is non-zero.
Correct answer is: No
Q.16 Which of the following is not a valid type of matrix?
Diagonal
Triangular
Hexagonal
Identity
Explanation - Hexagonal matrix is not a mathematical classification; others are standard forms.
Correct answer is: Hexagonal
Q.17 What is det([[0,1],[-1,0]])?
1
-1
0
2
Explanation - Det = (0*0 - 1*-1) = 1.
Correct answer is: 1
Q.18 The product of a matrix and its inverse is:
0
1
Identity matrix
Transpose
Explanation - By definition, A·A⁻¹ = I, the identity matrix.
Correct answer is: Identity matrix
Q.19 If two rows of a determinant are identical, the determinant is:
0
1
Unchanged
Negative
Explanation - If two rows or columns are identical, determinant = 0.
Correct answer is: 0
Q.20 What is det([[1,0,0],[0,1,0],[0,0,1]])?
1
3
0
9
Explanation - This is the 3x3 identity matrix; determinant = 1.
Correct answer is: 1
Q.21 If det(A) = 7, what is det(Aᵀ)?
-7
0
7
49
Explanation - Determinant of transpose equals determinant of the matrix.
Correct answer is: 7
Q.22 What is det([[2,3],[4,6]])?
0
12
6
1
Explanation - Det = (2*6 - 3*4) = 12 - 12 = 0.
Correct answer is: 0
Q.23 If A is an upper triangular matrix, det(A) is equal to:
Sum of diagonal
0
Product of diagonal
Rank
Explanation - For triangular matrices, determinant equals product of diagonal elements.
Correct answer is: Product of diagonal
Q.24 Which of the following is a null matrix?
[[0,0],[0,0]]
[[1,0],[0,1]]
[[2,3],[4,5]]
[[0,1],[1,0]]
Explanation - A null matrix has all entries equal to zero.
Correct answer is: [[0,0],[0,0]]
Q.25 The inverse of a matrix A exists if and only if:
det(A) = 0
det(A) ≠ 0
A is square
A is diagonal
Explanation - A matrix has an inverse if and only if it is non-singular, i.e., determinant ≠ 0.
Correct answer is: det(A) ≠ 0