Linear Algebra

Question 41

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

• row is the same as its second row
• row is the same as the second row of A
• column is the same as the second column of A
• row is all zero

Question 42

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 an n×1 column vector of unknowns. Which of the following is false?

• The system has a solution if and only if, both A and the augmented matrix [Ab] have the same rank
• If m
• If m=n and b is a non-zero vector, then the system has a unique solution
• The system will have only a trivial solution when m=n, b is the zero vector and rank(A) = n

Question 43

The matrices commute under multiplication

• if a = b or Θ = nπ, n an integer
• always
• never
• if a cosΘ = b sin Θ

Question 44

Let be two matrices such that Express the elements of D in terms of the elements of B.

Question 45

An orthogonal matrix A has eigen values 1, 2 and 4. What is the trace of the matrix [Tex]A^T[/Tex]?

• 7/4
• 1/7
• 7
• 4/7

Question 46

Let 

A = [Tex]\begin{bmatrix} 
4 & -2 & 1 \\ 
2 & 0 & 1 \\
2 & -2 & 3
\end{bmatrix}[/Tex]



v1 = [Tex]\begin{bmatrix} 
1  \\ 
1  \\
0 
\end{bmatrix}[/Tex]



v2 = [Tex]\begin{bmatrix} 
0  \\ 
1  \\ 
2 
\end{bmatrix}[/Tex]



v3 = [Tex]\begin{bmatrix} 
1  \\ 
2  \\ 
-1 
\end{bmatrix}[/Tex]



v4 = [Tex]\begin{bmatrix} 
1  \\ 
1  \\ 
4 
\end{bmatrix}[/Tex]

Which of the following statements is correct ? 

• v₃ and v₄ are eigenvectors of A
  

• v₂ and v₃ are eigenvectors of A
  

• v₁ and v₃ are eigenvectors of A
  

• v₁ and v₂ are eigenvectors of A
  

Question 47

The matrix A has (1, 2, 1)^T and (1, 1, 0)^T as eigenvectors, both with eigenvalue 7, and its trace is 2. The determinant of A is __________ .

• 84
• 588
• 49
• None of these

Question 48

Find the determinant of the following matrix:

• -708
• -452
• -844
• -588

Question 49

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 paranthesization if they are directly multiplied. For example, in the matrix multiplication chain G₁G₂G₃G₄G₅G₆ using parenthesization (G₁(G₂G₃))(G₄(G₅G₆)), G₂G₃ and G₅G₆ are 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

• F₁F₂ and F₃F₄ only
• F₂F₃ only
• F₃F₄ only
• F₁F₂ and F₄F₅ only

Question 50

Consider a matrix P 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?

• Only I and III are necessarily true
• Only II is necessarily true
• Only I and II are necessarily true
• Only II and III are necessarily true