If the following system of equations possess a $$non-trivial$$ solution over the set of rationals $$x+y$$−$$2z=0$$, $$2x$$−$$3y+z=0andx$$−$$5y+4z=k$$ $$thenk+11000=$$

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If the following system of equations possess a $$non-trivial$$ solution over the set of rationals $$x+y$$−$$2z=0$$, $$2x$$−$$3y+z=0andx$$−$$5y+4z=k$$ $$thenk+11000=$$

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Augmented matrix $$AD=11-202-3101-54k$$                                                apply $$R2$$→$$R2-2R1$$, $$R3$$→$$R3-R1$$                                             ~$$11-200-5500-66k$$                                                 apply  $$R2$$→$$R25$$,  $$R3$$→$$R36$$                                              ~$$11-200-1100-11k6$$                                                    apply $$R3$$→$$R3-R2$$                                              ~$$11-200-110000k6$$ Since the system possesses a $$non-trivial$$ solution, the system is consistent. ⇒$$rankA=rankAD$$ ⇒$$k6=0$$ ⇒$$k=0$$ ∴$$k+11000=11000=0.001$$

System of linear equations

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If the following system of equations possess a non-trivial solution over the set of rationals $x + y - 2 z = 0, 2 x - 3 y + z = 0$ and $x - 5 y + 4 z = k$ then $\frac{k + 1}{1000} =$

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System of linear equations

Remember concepts with our Masterclasses.

If the following system of equations possess a non-trivial solution over the set of rationals x+y−2z=0, 2x−3y+z=0andx−5y+4z=k thenk+11000=

Augmented matrix AD=11-202-3101-54k apply R2→R2-2R1, R3→R3-R1 ~11-200-5500-66k apply R2→R25, R3→R36 ~11-200-1100-11k6 apply R3→R3-R2 ~11-200-110000k6 Since the system possesses a non-trivial solution, the system is consistent. ⇒rankA=rankAD ⇒k6=0 ⇒k=0 ∴k+11000=11000=0.001

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If the following system of equations possess a non-trivial solution over the set of rationals $x + y - 2 z = 0, 2 x - 3 y + z = 0$ and $x - 5 y + 4 z = k$ then $\frac{k + 1}{1000} =$