suppose that AX=0 is homogeneous system of n equations in nvariables 1)if the sy
ID: 2939859 • Letter: S
Question
suppose that AX=0 is homogeneous system of n equations in nvariables 1)if the system A2X=0 has a nontrivial soltion,show that AX=0 also has a nontrivial solution hint- prove the contrapositive 2)generalize the result of part (1) to show that if the systemAnX=0 has a nontrivial solution for some positiveinterger n, then AX=0 has a nontrivial solution. hint- use a proof by indcution. thank you suppose that AX=0 is homogeneous system of n equations in nvariables 1)if the system A2X=0 has a nontrivial soltion,show that AX=0 also has a nontrivial solution hint- prove the contrapositive 2)generalize the result of part (1) to show that if the systemAnX=0 has a nontrivial solution for some positiveinterger n, then AX=0 has a nontrivial solution. hint- use a proof by indcution. thank youExplanation / Answer
suppose that AX=0 is homogeneous system of n equations in nvariables 1)if the system A2X=0 has a nontrivial soltion,show that AX=0 also has a nontrivial solution hint- prove the contrapositivethe system A2X=0 has a nontrivial solution
HENCE A2 IS SINGULAR MATRIX
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[ IN THE SET OF EQNS.... MX=0, IF M IS NON SINGULAR, THEN THEINVERSE OF M EXISTS
AND HENCE, WE GET ...M-1MX=M-1*0...THATIS..... IX=0.....OR.....X=0...THAT IS
WE GET THE TRIVIAL SOLUTION ...HENCE TO GET A NON TRIVIALSOLUTION.
M SHOULD BE SINGULAR]
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THAT IS
|A2|=|A||A|=0
HENCE
|A|=0
THAT IS A IS SINGULAR MATRIX
HENCE NON TRIVIAL SOLUTION EXISTS FOR THE SET OF EQNS....AX=0
2)generalize the result of part (1) to show that if the systemAnX=0 has a nontrivial solution for some positiveinterger n, then AX=0 has a nontrivial solution.
ARGUING IN THE SAME MANNER AS ABOVE , WE GET
|AN|=0
|A||A|A|.......N TIMES =0
|A|=0
THAT IS A IS SINGULAR MATRIX
HENCE NON TRIVIAL SOLUTION EXISTS FOR THE SET OF EQNS....AX=0
hint- use a proof by indcution. thank you
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