Consider the matrix  below:

For which of the following combinations of , and , is the rank of  at least three?

(i)  and .

(ii) .

(iii)  and .

(iv) .

Consider the vectors

For real-valued scalar variable , the value of

  is _________ (rounded off to two decimal places).

 denotes the Euclidean norm, i.e., for .

Let  and  denote the set of real numbers and the three-dimensional vector space over it, respectively. The value of for which the set of vectors

Dose not form a basis of is__________.

Consider the matrix where k is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

Let  and  be two vectors. The value of the coefficient  in the expression , which minimizes the length of the error vector , is

Let the sets of eigenvalues and eigenvectors of a matrix  be  and , respectively. For any invertible matrix , the sets of eigenvalues and eigenvectors of the matrix , where , respectively, are

Let  be an  real column vector with length . The trace of the matrix  is