Linear Algebra

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 G₁ G₂ G₃ … G_n can be done by parenthesizing in different ways.

Define G_i G_i+1 as an explicitly computed pair for a given parenthesization if they are directly multiplied.

For example, in the matrix multiplication chain G₁ G₂ G₃ G₄ G₅ G₆ using the parenthesization (G₁ (G₂ G₃))(G₄ (G₅ G₆)), the explicitly computed pairs are G₂ G₃ and G₅ G₆.

Consider a matrix multiplication chain F₁ F₂ F₃ F₄ F₅, where the dimensions of the matrices are:

• F1: 2 × 25
• F2: 25 × 3
• F3: 3 × 16
• F4: 16 × 1

In the parenthesization of F₁ F₂ F₃ F₄ F₅ that minimizes the total number of scalar multiplications, the explicitly computed pair(s) is/are:

Which of the following is used to determine the specificity of requirements? (1) n₁ / n₂ (2) n₂ / n₁ (3) n₁ + n₂ (4) n₁ - n₂ Where n₁ is the number of requirements for which all reviewers have identical interpretations, n₂ is number of requirements in a specification.

A neuron with 3 inputs has the weight vector [0.2 –0.1 0.1]^T and a bias θ = 0. If the input vector is X = [0.2 0.4 0.2]^T then the total input to the neuron is :

Let R and S be two fuzzy relations defined as : Then, the resulting relation, T, which relates elements of universe x to the elements of universe z using max-min composition is given by :

If A is a skew symmetric matrix, then A^t

A 4×4 DFT matrix is given by :

[Tex]\frac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & x & -1 & y \\ 1 & -1 & 1 & -1 \\ 1 & -j & -1 & j \end{bmatrix}[/Tex]

(j² = −1)
Where values of x and y are _____, _____ respectively.

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

If C is a skew-symmetric matrix of order n and X is n x 1 column matrix, then X^TCX is a

Find the boolean product A⊙B of the two matrices.

Consider a matrix P(2x2) whose only eigenvectors are the multiples of [Tex]\begin{bmatrix}1 \\ 4 \end{bmatrix}[/Tex]
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?