Answer to Question #226355 in Linear Algebra for Nikhil Singh

Question

a. The relation ‘~' on Z given by (a,b)~(c,d)⟺ (a-b)| (c-d) is an equivalence relation.
True or false with full explanation

b. A 3×3 matrix of rank one has an eigenvalue zero.
True or false with full explanation

Answer 1

Solution:

(a): Given: A relation ~ on Z given by (a,b)~(c,d)⟺ (a-b)| (c-d)

Reflexive: (a-b)|(a-b) $\forall$ a,b$\in$ Z.

So, (a,b)~(a,b) holds,

Thus, ~ is reflexive.

Symmetric: Let a,b,c,d $\in$ Z such that (a-b)|(c-d) holds.

$\Rightarrow$ (a-b)|(c-d) but it does not imply that (c-d)|(a-b).

For example: a=10, b=8, c=9, d=3

Then, a-b = 10-8=2, c-d=9-3=6

Now, 2|6 but 6 does not divide 2.

Thus, ~ is not symmetric.

Thus, ~ is not an equivalence relation.

So, the given statement is False.

(b): Given a 3x3 matrix has rank one. So, the matrix is singular.

So, the determinant of this matrix is 0.

We know that determinant of a matrix = Product of its eigenvalues

So, 0 must be an eigenvalue of this matrix,

Thus, the given statement is True.