Rank of Matrix and Digonalization

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Question 1

The rank of the given matrix is:

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

  • 1
     

  • 0
     

  • 2
     

  • 4
     

Question 2

The rank of the matrix given below is:

[Tex]\begin{bmatrix}1 & 4 & 8 & 7 \\0 & 0 & 3 & 0 \\4 & 2 & 3 & 1 \\3 & 12 & 24 & 21 \\\end{bmatrix}[/Tex]

  • 3

  • 1

  • 2

  • 4

Question 3

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?

  • I and II only

  • I and IV only

  • II and III only

  • III and IV only

Question 4

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?

  • 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

Question 5

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

  • i + j

  • (j - 1) + i(i - 1)/2

  • i + j - 1

  • i + j(j - 1)/2

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