Q.1 If A = [[1, 2], [3, 4]], what is det(A)?
-2
-1
2
1
Explanation - For a 2x2 matrix [[a,b],[c,d]], determinant is ad - bc. Here, det(A) = (1*4)-(2*3) = 4-6 = -2.
Correct answer is: -2
Q.2 If A = [[1, 0], [0, 1]], then A is called?
Zero matrix
Identity matrix
Diagonal matrix
Singular matrix
Explanation - A square matrix with 1s on the diagonal and 0s elsewhere is called the identity matrix.
Correct answer is: Identity matrix
Q.3 The sum of two matrices is defined if:
They are square
They are of same order
They are invertible
They are diagonal
Explanation - Two matrices can be added only if they have the same dimensions (same number of rows and columns).
Correct answer is: They are of same order
Q.4 If det(A) = 0, then A is called?
Invertible
Singular
Diagonal
Symmetric
Explanation - A matrix whose determinant is 0 is called singular and it does not have an inverse.
Correct answer is: Singular
Q.5 If A = [[2,3],[1,4]], what is the inverse of A?
[[4,-3],[-1,2]]
[[2,3],[1,4]]
[[0.8,-0.6],[-0.2,0.4]]
[[1,0],[0,1]]
Explanation - Inverse of 2x2 matrix [[a,b],[c,d]] is (1/det) * [[d,-b],[-c,a]]. Here det = 2*4-3*1 = 5, so inverse = (1/5)*[[4,-3],[-1,2]] = [[0.8,-0.6],[-0.2,0.4]].
Correct answer is: [[0.8,-0.6],[-0.2,0.4]]
Q.6 The determinant of a triangular matrix is:
Sum of diagonal elements
Product of diagonal elements
Always zero
Always one
Explanation - For a triangular (upper or lower) matrix, the determinant is the product of its diagonal entries.
Correct answer is: Product of diagonal elements
Q.7 If A is a 3x3 matrix and |A| = 5, what is |2A|?
10
40
8
40
Explanation - For an n x n matrix, |kA| = k^n * |A|. Here, n=3, k=2, so |2A| = 2^3 * 5 = 8*5=40.
Correct answer is: 40
Q.8 If A and B are square matrices of same order, then det(AB) = ?
det(A) + det(B)
det(A) * det(B)
det(A) - det(B)
det(A)/det(B)
Explanation - Determinant of the product of two matrices equals the product of their determinants.
Correct answer is: det(A) * det(B)
Q.9 Which of the following is always true for any square matrix A?
det(A^T) = det(A)
det(A^T) = -det(A)
det(A^2) = det(A)
det(2A) = 2*det(A)
Explanation - The determinant of a square matrix is equal to the determinant of its transpose.
Correct answer is: det(A^T) = det(A)
Q.10 If a 3x3 matrix has two identical rows, then its determinant is:
0
1
2
Undefined
Explanation - A determinant is zero if any two rows or columns of the matrix are identical.
Correct answer is: 0
Q.11 For matrix A = [[1,2,3],[0,1,4],[5,6,0]], what is det(A)?
1
24
-1
0
Explanation - Using determinant formula for 3x3: det(A) = 1*(1*0-4*6) - 2*(0*0-4*5) + 3*(0*6-1*5) = 1*(-24) -2*(-20)+3*(-5)=-24+40-15=1.
Correct answer is: 1
Q.12 The matrix [[0,1],[-1,0]] represents a:
Reflection matrix
Rotation matrix
Identity matrix
Zero matrix
Explanation - This 2x2 matrix represents rotation by 90 degrees counterclockwise in 2D.
Correct answer is: Rotation matrix
Q.13 If A is a 2x2 matrix with det(A) = 1, then the area of the parallelogram formed by column vectors of A is:
1
2
0
Cannot be determined
Explanation - The determinant of a 2x2 matrix gives the area of the parallelogram formed by its column vectors.
Correct answer is: 1
Q.14 Which of the following is not a property of determinants?
Interchanging two rows changes the sign
Multiplying a row by k multiplies det by k
Adding a multiple of one row to another row changes det
Determinant of identity matrix is 1
Explanation - Adding a multiple of one row to another does not change the determinant.
Correct answer is: Adding a multiple of one row to another row changes det
Q.15 A square matrix A is called skew-symmetric if:
A^T = A
A^T = -A
det(A)=0
A is diagonal
Explanation - A matrix is skew-symmetric if its transpose equals its negative.
Correct answer is: A^T = -A
Q.16 If A = [[1,2,3],[4,5,6],[7,8,9]], then det(A) = ?
0
1
45
-1
Explanation - The determinant is zero because the rows of A are linearly dependent.
Correct answer is: 0
Q.17 If A and B are invertible matrices, then (AB)^-1 = ?
A^-1 * B^-1
B^-1 * A^-1
A * B
B * A
Explanation - The inverse of a product is the product of inverses in reverse order.
Correct answer is: B^-1 * A^-1
Q.18 If det(A) = 3, what is det(A^-1)?
1/3
3
-3
-1/3
Explanation - The determinant of the inverse of a matrix is the reciprocal of the determinant.
Correct answer is: 1/3
Q.19 Which matrix is non-invertible?
[[1,2],[2,4]]
[[2,0],[0,2]]
[[1,0],[0,1]]
[[0,1],[-1,0]]
Explanation - This matrix has det = 1*4 - 2*2 = 0, so it is singular (non-invertible).
Correct answer is: [[1,2],[2,4]]
Q.20 The cofactor of an element a_ij in a matrix is:
(-1)^(i+j) * minor of a_ij
Sum of row i and column j
Product of row i and column j
Determinant of matrix without row i and column j
Explanation - Cofactor is signed minor: C_ij = (-1)^(i+j) * M_ij.
Correct answer is: (-1)^(i+j) * minor of a_ij
Q.21 Which of the following matrices is symmetric?
[[1,2],[2,1]]
[[1,2],[3,4]]
[[0,1],[-1,0]]
[[1,0],[0,-1]]
Explanation - A symmetric matrix satisfies A^T = A. Here the transpose equals the matrix itself.
Correct answer is: [[1,2],[2,1]]
Q.22 The sum of diagonal elements of a square matrix is called:
Trace
Determinant
Rank
Inverse
Explanation - The trace of a square matrix is the sum of its diagonal elements.
Correct answer is: Trace
Q.23 If A is 2x2 with det(A) = -5, then |3A| = ?
-45
-15
15
45
Explanation - For 2x2, |kA| = k^2 * |A|. So |3A| = 3^2 * (-5) = 9 * (-5) = -45. Correction: yes, answer = -45.
Correct answer is: 45
Q.24 The determinant of a 1x1 matrix [[a]] is:
0
a
1
Undefined
Explanation - Determinant of 1x1 matrix is the single element itself.
Correct answer is: a
Q.25 If det(A) = 7 and det(B) = 3, then det(A^2 * B) = ?
147
21
10
49
Explanation - det(A^2 * B) = det(A^2) * det(B) = (det(A))^2 * det(B) = 7^2 * 3 = 49 * 3 = 147.
Correct answer is: 147