If the matrix A⁴, calculated by the use of Cayley-Hamilton theorem or otherwise, is _________

The eigen vector(s) of the matrix

is (are)

Find the inverse of the matrix

The tank of matrix is:

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:

Let A and B be real symmetric matrices of size n × n. Then which one of the following is true?

Let A be the set of all non-singular matrices over real number and let* be the matrix multiplication operation. Then

In the interval [0, π] the equation x = cos x has

The rank of the following (n + 1)×(n+1) matrix, where a is a real number is

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?

Let and be two matrices such that AB = I. Let and CD = 1. Express the elements of D in terms of the elements of B.

The matrices and commute under multiplication

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 unknowns. Which of the following is false?

Consider the following matrix:

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

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

The determinant of the matrix is is:

Derive the expression for the number of expressions required to solve a system of linear equations in n unknowns using the Gaussian Elimination Method. Assume that one operation refers to a multiplication followed by an addition.

The rank of the matrix given below is:

        1   4   8   7
        0   0   3   0
        4   2   3   1
        3   12  24  2  

Consider the following determinant 

Which of the following is a factor of Δ?

Consider the following set of equations

                x + 2y = 5
               4x + 8y = 12
          3x + 6y + 3z = 15 

This set

The determinant of the matrix is

is:

An n x n array v is defined as follows:

v[i,j] = i-j for all i,j, 1 ≤ i ≤ n, 1 ≤ j ≤ n 

The sum of the elements of the array v is

Consider the following statements:

S1: The sum of two singular n × n matrices may be non-singular
S2: The sum of two n × n non-singular matrices may be singular.

Which of the following statements is correct?

(a) Obtain the eigen values of the matrix

(b) Determine whether each of the following is a tautology, a contradiction, or neither (“∨” is disjunction, “∧” is conjunction, “→” is implication, “¬” in negation, and “↔” is biconditional (if and only if).

   (i) A ↔ (A ∨ A)
   (ii) (A ∨ B) → B
   (iii) A ∧(¬(A ∨ B)

The rank of the matrix is

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 _______ .

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

The largest eigenvalue of A is ________

The number of integer-triples (i.j.k) with 1 ≤ i.j.k ≤ 300 such that i + j + k is divisible by 3 is ________