InterviewSolution
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InterviewSolution
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आव्यूह `[{:(,0,1,1),(,1,0,1),(,1,1,0):}]` का प्रतिलोम ज्ञात कीजिएः तथा दर्शाइए की `SAS^(-1)` एक विकर्ण आव्यूह है जहाँ `[{:(,b+c,c-a,b-a),(,c-b,c+a,a-b),(,b-c,a-c,a+b):}]` |
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Answer» यहाँ ` S= [ {:( 0,,1,,1), ( 1,,0,1) , ( 1,,1,,0) :}] ` ` therefore |S| = | {:(0 , 1, 1 ) , ( 1, 0 ,1) ,( 1, 1, 0 ) :} | ` ` rArr |S| = - 1 ( 0 - 1 ) + 1( 1 - 0 ) ` , [ ` R _ 1 ` के अनुदिश प्रसार करने पर ] ` rArr |S| = 2 ne 0 ` ` rArr S^( - 1 ) ` का अस्तित्व है | माना S के अवयवों का सहखंड ` S_ (ij ) ` है , तब ` S _ ( 11 ) = ( - 1 ) ^( 1 + 1 ) | {:(0,1) ,(1,0):}| = ( 0 - 1 ) = - 1 ` ` S _ ( 12 ) = ( - 1 ) ^ ( 1 + 2 ) |{:( 1 , 1 ) , ( 1 , 0 ) :}| = - ( 0 - 1 ) = 1 ` ` S_ ( 13 ) = ( - 1 ) ^( 1 + 3 ) |{:( 1 , 0 ) , ( 1, 1 ) :}| = ( 1 - 0 ) = 1 ` ` S _ ( 21 ) = ( - 1 ) ^( 2 + 1 ) |{:( 1, 1 ) ,( 1, 0 ) :} | = - ( 0 -1 ) = 1 ` ` S _ ( 22 ) = ( - 1 ) ^( 2 + 2 ) |{:( 0,1) , ( 1, 0 ) :}| = ( 0 - 1 ) = - 1 ` ` S _ ( 23 ) = ( - 1 ) ^ ( 2 + 3 ) |{:( 0 , 1 ) , ( 1, 1 ) :}| = - ( 0 - 1 ) = - 1 ` ` S _ ( 31 ) = ( - 1 ) ^( 3 + 1 ) |{:( 1 , 1 ) , ( 0 , 1 ) :}| = ( 1 - 0 ) = 1 ` ` S _ ( 32 ) = ( - 1 ) ^( 3 + 2 ) |{:( 0 , 1 ) ,( 1, 1 ) :} | = - ( 0 - 1 ) = 1 ` ` S_ ( 33 ) = ( - 1 ) ^( 3 + 3 ) |{:( 0 , 1 ) , ( 1, 0 ) :}| = ( 0 - 1 ) = - 1 ` ` therefore adj S = [{:( - 1, 1, 1 ) , ( 1, - 1 , 1 ) , ( 1, 1 , - 1 ) :}] ` अतः ` S^( - 1 ) = ( adj S ) /( |S| ) = ( 1 ) /( 2 ) [{:( - 1, 1, 1 ) , ( 1 , - 1, 1 ) ,( 1, 1 , - 1 ) :}] ` अब `SA = [{:( 0 , 1, 1 ) , ( 1, 0 , 1 ) , ( 1, 1, 0 ):}] xx ( 1 ) /( 2 ) [{:( b + c, c - a, b - a ) ,( c - b , c + a , a - b ) , ( b - c , a - c , a + b ) :}] ` ` = ( 1 ) /( 2) [{:( 0 + c- b + b - c ,, 0 + ( c + a ) + ( a - c ) ,, 0 + ( a - b ) + ( a + b ) ), ( ( b + c ) + 0 + ( b - c ) ,, ( c - a ) + 0 + ( a - c ) ,, ( b - a ) + 0 + ( a + b )) , ( ( b + c ) + ( c - d ) + 0 ,, ( c - a ) + ( c + a ) + 0 ,, ( b - a ) + ( a - b ) + 0 ) :}]` ` = ( 1 ) /( 2 ) [{:( 0 , 2a , 2a ) , ( 2b, 0 , 2b ) , ( 2c, 2c , 0 ) :}] = [{:( 0 , a, a ) , ( b, 0 , b ) , ( c, c, 0 ):}] ` ` therefore SAS^( - 1 ) = ( 1 ) /( 2 ) xx [ {:( 0 , a , a ) , ( b , 0 , b ), ( c , c , 0 ) :}][{:( - 1,1, 1) , ( 1, - 1 , 1 ) , ( 1, 1 , - 1 ) :}] ` ` = ( 1 ) /( 2 ) [{:( 0 + a + a , 0 - a + a , 0 + a - a ) , ( - b + 0 + b , b + 0 + b , b + 0 - b ) , ( - c + c + 0 , c - c + 0 , c + c - 0 ) :}] ` ` = ( 1 ) /( 2 ) [{:( 2a, 0 , 0 ) ,( 0 , 2b , 0 ) , ( 0 , 0 , 2c ) :}] ` ` = [{:( a, 0 , 0 ) , ( 0 , b , 0 ) , ( 0 , 0 , c ) :} ` ` rArr SAS^( - 1 ) ` एक विकर्ण आव्यूह है | यही सिद्ध करना था | |
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