Assume that multiplying a matrix G₁ of dimension p×q with another matrix G₂ of dimension q×r requires pqr scalar multiplications. Computing the product of n matrices G₁G₂G₃…G_n can be done by parenthesizing in different ways. Define G_iG_i+1 as an explicitly computed pair for a given parenthesization if they are directly multiplied. For example, in the matrix multiplication chain G₁G₂G₃G₄G₅G₆ using parenthesization(G₁(G₂G₃))(G₄(G₅G₆)), G₂G₃ and G₅G₆ are the only explicitly computed pairs.

Consider a matrix multiplication chain F₁F₂F₃F₄F₅, where matrices F₁, F₂, F₃, F₄ and F₅ are of dimensions 2×25, 25×3, 3×16, 16×1 and 1×1000, respectively. In the parenthesization of F₁F₂F₃F₄F₅ that minimizes the total number of scalar multiplications, the explicitly computed pairs is/ are

Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium (M) and low (L). Let P(H_G) denote the probability that Guwahati has high temperature. Similarly, P(M_G) and P(L_G) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P(H_D), P(M_D) and P(L_D) for Delhi.

The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature.

Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature (H_G) then the probability of Delhi also having a high temperature (H_D) is 0.40; i.e., P(H_D ∣ H_G) = 0.40. Similarly, the next two entries are P(M_D ∣ H_G) = 0.48 and P(L_D ∣ H_G) = 0.12. Similarly for the other rows.

If it is known that P(H_G) = 0.2, P(M_G) = 0.5, and P(L_G) = 0.3, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _______ .

Let N be the set of natural numbers. Consider the following sets,

P: Set of Rational numbers (positive and negative)
Q: Set of functions from {0, 1} to N
R: Set of functions from N to {0, 1}
S: Set of finite subsets of N

Which of the above sets are countable?

Consider the following statements.

(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized

Which one of the following options is correct?

Let u and v be two vectors in R² whose Euclidean norms satisfy ||u||=2||v||. What is the value of α such that w = u + αv bisects the angle between u and v?

(i)  One eigenvalue must be in [-5, 5].
(ii) The eigenvalue with the largest magnitude 
     must be strictly greater than 5.

If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ³ - 4λ² + aλ + 30, a ∈ ℝ, and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is ________.

Two eigenvalues of a 3 × 3 real matrix P are (2 + √-1) and 3. The determinant of P is __________.

Consider the systems, each consisting of m linear equations in n variables.

I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution

Which one of the following is CORRECT?

Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (A^-1)^T is _________.

| 2  2 |
| 4  9 |

A = | 1  4 |
    | b  a |

⎡ 1 -1  2 ⎤
⎢ 0  1  0 ⎥
⎣ 1  2  1 ⎦

  px + qy + rz = 0
  qx + ry + pz = 0
  rx + py + qz = 0

1 0 0 0 1
0 1 1 1 0
0 1 1 1 0
0 1 1 1 0
1 0 0 0 1

[A  I]
[I  A]

Consider the following system of linear equation

Let a = (a_ij) be an n-rowed square matrix and I₁₂ be the matrix obtained by interchanging the first and second rows of the n-rowed Identify matrix. Then AI₁₂ is such that its first

Let Ax = b be a system of linear equations where A is an m × n matrix and b is a m × 1 column vector and X is a n × 1 column vector of unknows. Which of the following is false?

In a compact single dimensional array representation for lower triangular matrices (i.e all the elements above the diagonal are zero) of size n × n, non-zero elements (i.e elements of the lower triangle) of each row are stored one after another, starting from the first row, the index of the (i, j)^th element of the lower triangular matrix in this new representation is: