GATE DA Linear Algebra Quiz

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Question 1

[Tex]\text{Let } M = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6 \end{bmatrix}[/Tex]. [Tex]\text{Find } \det(M^2 + 12M)[/Tex]

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Question 2

Let D = {x(1), . . . , x(n)} be a dataset of n observations where each x(i) ∈ R100. It is given that [Tex]\sum\limits^{n}_{i=1}x^{(i)}=0[/Tex]. The covariance matrix computed from D has eigenvalues Ξ»i = 1002βˆ’i, 1 ≀ i ≀ 100. Let u ∈ R100 be the direction of maximum variance with uTu = 1.

The value of

1

(Answer in integer)

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  • 100

  • 300

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Question 3

Consider the matrix:

Matrix


Which ONE of the following statements is TRUE?

  • The eigenvalues of M are non-negative and real.

  • The eigenvalues of M are complex and conjugate pairs.

  • One eigenvalue of is positive and real and another eigen value of M is Zero.

  • One eigenvalue of M is non-negative and real,and another eigen value of M is negative and real.

Question 4

Select all choices that are subspaces of ℝ 3 .

Note: ℝ denotes the set of real numbers.

  • [Tex]\left\{\mathbf{x} =\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}\in \mathbb{R}^3 :\mathbf{x} = \alpha\begin{bmatrix}1 \\1 \\0\end{bmatrix}+ \beta\begin{bmatrix}1 \\0 \\0\end{bmatrix}, \, \alpha, \beta \in \mathbb{R}\right\}[/Tex]

  • [Tex]\left\{\mathbf{x} =\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}\in \mathbb{R}^3 :\mathbf{x} = \alpha^2\begin{bmatrix}1 \\2 \\0\end{bmatrix}+ \beta^2\begin{bmatrix}1 \\0 \\1\end{bmatrix}, \, \alpha, \beta \in \mathbb{R}\right\}[/Tex]

  • [Tex]\left\{\mathbf{x} =\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}\in \mathbb{R}^3 :5x_1 + 2x_3 = 0, \, 4x_1 - 2x_2 + 3x_3 = 0\right\}[/Tex]

  • [Tex]\left\{\mathbf{x} =\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}\in \mathbb{R}^3 :5x_1 + 2x_3 + 4 = 0\right\}[/Tex]

Question 5

Which of the following statements is/are TRUE?

Note: ℝ denotes the set of real numbers.

  • There exist 𝑴 ∈ ℝ3Γ—3 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has a unique solution and M𝐱 = 𝒒 has infinite solutions.

  • There exist 𝑴 ∈ ℝ3Γ—3 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has no solution and M𝐱 = 𝒒 has infinite solutions.

  • There exist 𝑴 ∈ ℝ2Γ—3 , 𝒑 ∈ ℝ2 , and 𝒒 ∈ ℝ2 such that 𝑴𝐱 = 𝒑 has a unique solution and M𝐱 = 𝒒 has infinite solutions.

  • There exist 𝑴 ∈ ℝ3Γ—2 , 𝒑 ∈ ℝ3 , and 𝒒 ∈ ℝ3 such that 𝑴𝐱 = 𝒑 has a unique solution and M𝐱 = 𝒒 has no solution.

Question 6

Let ℝ be the set of real numbers, π‘ˆ be a subspace of ℝ3 and 𝑴 ∈ ℝ3Γ—3 be the matrix corresponding to the projection on to the subspace π‘ˆ. Which of the following statements is/are TRUE?

  • If π‘ˆ is a 1-dimensional subspace of ℝ3 , then the null space of 𝑴 is a 1-dimensional subspace.

  • If π‘ˆ is a 2-dimensional subspace of ℝ3 , then the null space of 𝑴 is a 1-dimensional subspace.

  • M2=M

  • M3=M

Question 7

[Tex][/Tex]

[Tex]\text{Let } \mathbf{u} = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}[/Tex]

, and let [Tex]\sigma_1, \; \sigma_2, \; \sigma_3, \; \sigma_4, \; \sigma_5[/Tex] be the singular values of the matrix [Tex]M = uu^{T}[/Tex] (where [Tex]u^T[/Tex] is the transpose of 𝒖). The value of [Tex]\sum_{i=1}^{5} \sigma_i[/Tex] is ______.

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Question 8

The number of additions and multiplications involved in performing Gaussian elimination on any n Γ— n upper triangular matrix is of the order

  • O(n)

  • O(n2)

  • O(n3)

  • O(n4)

Question 9

The sum of the elements in each row of A ∈ [Tex]R^{n \times n}[/Tex] is 1. If [Tex]B = A^3 βˆ’ 2A^2 + A[/Tex], which one of the following statements is correct (for x ∈ Rn )? 

  • The equation Bx = 0 has no solution

  • The equation Bx = 0 has exactly two solutions

  • The equation Bx = 0 has infinitely many solutions

  • The equation Bx = 0 has a unique solution

Question 10

Which of the following statements is/are correct?

  • Rn has a unique set of orthonormal basis vectors

  • Rn does not have a unique set of orthonormal basis vectors

  • Linearly independent vectors in Rn are orthonormal

  • Orthonormal vectors Rn are linearly independent

There are 15 questions to complete.

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