Linear Algebra

Question 71

If the two matrices have the same determinant, then the value of x is

• 1/2
• √2
• ± 1/2
• ± 1/√2

Question 72

Consider the matrix M = representing a set of planar (2D) geometric transformations in homogeneous coordinates. Which of the following statements about the matrix M is True?

• M represents first, a scaling of vector (2, 1) followed by translation of vector (1, 1)
• M represents first, a translation of vector (1, 1) followed by scaling of vector (2, 1)
• M represents first, a scaling of vector (3, 1) followed by shearing of parameters (−1, 1)
• M represents first, a shearing of parameters (−1, 1) followed by scaling of vector (3, 1)

Question 73

Let A and B be two n×n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB) = rank(A)*rank (B)
II. det(AB) = det(A)*det(B)
III. rank(A+B) ≤ rank(A) + rank(B)
IV. det(A+B) ≤ det(A) + det(B) 

Which of the above statements are TRUE ?

• I and II only
• I and IV only
• II and III only
• III and IV only

Question 74

Consider the following matrix. The largest eigenvalue of the above matrix is __________.

• 3
• 4
• 0
• 12

Question 75

Suppose that P is a 4×5 matrix such that every solution of the equation Px=0 is a scalar multiple of [2 5 4 3 1]^T. The rank of P is __________ .

• 4
• 5
• 1
• 2

Question 76

For two n-dimensional real vectors P and Q, the operation s(P,Q) is defined as follows: Let L be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors P,Q∈L, s(P,Q)=0. What is the maximum cardinality possible for the set L?

• 9
• 10
• 11
• 100

Question 77

If log(base a) 2 = x and log(base a) 8 = y, what is the value of log(base a) 16? 

• 2x + y 

• 2x - y 

• x + y 

• 4x