Answer to Question #162104 in Linear Algebra for Sourav Mondal

Question

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.

Answer 1

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 , $c\in Q$ and $c\ne 0$ . Then D₂=c^-1D₁ and, hence, D₂~D₁.

(c) Transitivity: D₁~D₂ and D₂~D₃ imply D₁ = c₁D₂ and D₂ = c₂D₃ with $c_1,c_2\in Q\setminus\{0\}$ . Then D₁ = cD₃ with $c=c_1c_2\ne 0$ . 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, $A=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ 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, $A=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$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.