Which of the following statements are true ?
Give reasons for your answers in the form of a
short proof or a counter-example : 10
(1) If X={D|D is a 2 x 2 diagonal matrix) and
D1~D2 iff D1 = cD2, c not equal to 0, c E Z' , then~ is an equivalence relation on X.
(ii) If two matrices A and B have the same
eigen values, then A = B.
(iii) Any symmetric matrix is non-singular.
(iv) If V = {(x, y) belongs to R² |x = y} , then (1, 0) +V and (0, 1) + V are two distinct elements of
R2/V.
1. D1~D2 iff D1 = cD2, c not equal to 0, c E Z' , then~ is an equivalence relation on X. If Z' means the field of partials of the ring Z (that is, Q), then the statement is true. But if Z' means Z, then the statement is false, because of in this case the relation ~ is not symmetrical: (D~2D, but not 2D~D). Let Z'=Q now.
(a) Reflexivity: D~D (=1D) - is satisfied
(b) Symmetry: D1~D2 implies D1 = cD2 , and . Then D2=c-1D1 and, hence, D2~D1.
(c) Transitivity: D1~D2 and D2~D3 imply D1 = c1D2 and D2 = c2D3 with . Then D1 = cD3 with . Therefore D1~D3 and the property of Transitivity is satisfied.
Conclusion. The relation ~ is an equivalence relation.
2. If two matrices A and B have the same eigen values, then not necessary that A=B. For example, and have the same eigen values (0 and 0), but not equal.
3. The statement that any symmetric matrix is non-singular is false. For example, is symmetric, but singular.
4. The statement that "if V = {(x, y) belongs to R² |x = y} , then (1, 0) +V and (0, 1) + V are two distinct elements of R2/V" is true, becase of the difference between (1,0) and (0,1) equals (1,-1) and it does not belong to V.
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