Consider solving the following system of simultaneous equations using LU decomposition. 

x₁ + x₂ - 2x₃ = 4

x₁ + 3x₂ - x₃ = 7

2x₁ + x₂ - 5x₃ = 7

where L and U are denoted as

Which one of the following is the correct combination of values for L₃₂, U₃₃, and x₁?   [ MCQ || GATE 2022 || 2 Marks ]

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: [ MCQ || GATE 2018 || 2 Marks]

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? [ MCQ || GATE 2018 || 2 Marks]

[ NAT || GATE 2018 || 1 Marks]

Let X be a square matrix. Consider the following two statements on X.

      I. X is invertible.

      II. Determinant of X is non-zero.

Which one of the following is TRUE? [ MCQ || GATE 2019 || 1 Marks]

Let U = {1, 2, …,n} and A = {(x, X)∣ x∈X, X⊆U }. Consider the following two statements on |A|.

Which of the above statements is/are TRUE? [ MCQ || GATE 2019 || 2 Marks]

1.

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? [MCQ || GATE 2020 || 2 Marks]

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 _______ . [ NAT || GATE 2021 Set-2 || 1 Marks]

Consider the following matrix.

The largest eigenvalue of the above matrix is __________.[ NAT || GATE 2021 Set - 1 || 2 Marks]