Consider the following matrix:

The absolute value of the product of Eigen values of R is ______.

The largest eigenvalue of A is ________

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 c¹, cⁿ be scalars not all zero. Such that the following expression holds:

where a_i is column vectors in Rⁿ. Consider the set of linear equations.

where A = [a₁.......a_n] and

Then, Set of equations has

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?

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 __________.

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

In the LU decomposition of the matrix ,if the diagonal elements of U are both 1, then the lower diagonal entry l₂₂ of L is

Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are –1 and 7. What are the values of a and b?

The larger of the two eigenvalues of the matrix is _________.

Perform the following operations on the matrix

(i) add the third row to the second row
(ii) Subtract the third column from the first column

The determinant of the resultant matrix is ________.

In the given matrix, one of the eigenvalues is 1. the eigenvectors corresponding to the eigenvalue 1 are

⎡ 1 -1  2 ⎤
⎢ 0  1  0 ⎥
⎣ 1  2  1 ⎦

If the following system has non-trivial solution,

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

then which one of the following options is True?

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is ________.

If the matrix A is such that

then the determinant of A is equal to

The product of the non-zero eigenvalues of the matrix

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

is ______.

Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?

Which one of the following does NOT equal to

Let A be the 2×2 matrix with elements a₁₁ = a₁₂ = a₂₁ = +1 and a₂₂ = -1. Then the eigenvalues of the matrix A¹⁹ are

Consider the matrix as given below.

Which one of the following provides the CORRECT values of eigenvalues of the matrix?

Consider the following matrix . If the eigenvalues of A are 4 and 8, then

The following system of equations

x₁ + x₂ + 2x₃  = 1
x₁ + 2x₂ + 3x₃ = 2
x₁ + 4x₂ + ax₃ = 4

has a unique solution. The only possible value(s) for a is/are

How many of the following matrices have an eigenvalue 1?

Let A be a 4 x 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of

[A  I]
[I  A]

where I is the 4 x 4 identity matrix?

Consider the set of (column) vectors defined by X = {x ∈ R³| x₁+x₂+x₃ = 0, where x^T = [x₁,x₂,x₃]^T}. Which of the following is TRUE?

F is an n×n real matrix. b is an n×1 real vector. Suppose there are two n×1 vectors, u and v such that, u≠v and Fu=b, Fv=b. Which one of the following statements is false?

Consider the following system of equations in three real variables x_l, x₂ and x₃

2xl - x2 + 3x3 = 1 
3xl- 2x2 + 5x3 = 2 
-xl + 4x2 + x3 = 3 

This system of equations has

What are the eigenvalues of the following 2 × 2 matrix?

|2   -1|
|-4   5| 

The number of different n × n symmetric matrices with each element being either 0 or 1 is: (Note: power(2,x) is same as 2^x)

Let A, B, C, D be n × n matrices, each with non-­zero determinant. If ABCD = 1, then B^-1 is

How many solutions does the following system of linear equations have?

  -x + 5y = -1
   x - y = 2
   x + 3y = 3