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Show that {:[( 1+a,1,1),( 1,1+b,1) ,( 1...

Show that ` {:[( 1+a,1,1),( 1,1+b,1) ,( 1,1,1+c) ]:}= abc ( 1+(1)/(a) +(1)/(b) +(1)/(c) ) = abc +bc +ca +ab `

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Prove that |(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc(1+(1)/(a)+(1)/(b)+(1)/(c)).

Prove that : |{:(1+a, 1,1),(1,1+b,1),(1,1,1+c):}| =abc (1+1/a + 1/b + 1/c)

Prove that |(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc(1+1/a+1/b+1/c)

Evaluate Delta = {:[ (1,a,bc),( 1,b,ca),(1,c,ab)]:}

Prove that |{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|=(a-b)(b-c)(c-a) .

By using properties of determinants , show that : (i) {:[( 1,a,a^(2)),( 1,b,b^(2)),( 1,c,c^(2))]:}=(a-b)(b-c) (c-a) (ii) {:[( 1,1,1),( a,b,c) ,(a^(3) , b^(3), c^(3))]:} =( a-b) (b-c)( c-a) (a+b+c)

Prove that |{:(1,,ab,,(1)/(a)+(1)/(b)),(1 ,,bc,,(1)/(b)+(1)/(c)),( 1,, ca,,(1)/(c)+(1)/(a)):}|=0

If (1)/(a)+(1)/(c )=(1)/(2b-a)+(1)/(2b-c) , then

Prove that |(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2))|=(a-b)(b-c)(c-a)

Prove that (1)/(a) + (1)/(b) + (1)/(c ) ge (1)/(sqrt((bc))) + (1)/(sqrt((ca))) + (1)/(sqrt((ab))) , where a,b,c gt 0

NCERT TAMIL-DETERMINANTS -Miscellaneous Exercises on Chapter 4
  1. Show that {:[( 1+a,1,1),( 1,1+b,1) ,( 1,1,1+c) ]:}= abc ( 1+(1)/(a) +...

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  2. Prove that the determinant {:[( x, sin theta ,cos theta ),( -sin thet...

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  3. Without expanding the determinant, prove that {:[( a, a ^(2), bc ),( ...

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  4. Evaluate {:[( cos alpha cos beta , cos alpha sin beta , -sin alpha ),...

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  5. If a,b and c are real numbers, and Delta ={:[( b+c,C+a,a+b),( c+a,a...

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  6. Solve the equation {:[( x+a,x,x),(x,x+a,x),(x,x,x+a) ]:}=0,ane 0

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  7. Prove that {:[( a^(2) , bc, ac+c^(2)),( a^(2) +ab,b^(2) ,ac),( ab,b^(2...

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  8. If A^(-1) ={:[( 3,-1,1),(-15,6,-5),(5,-2,2) ]:}and B= {:[( 1,2,-2),( -...

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  9. Let A = {:[( 1,2,1),(2,3,1),(1,1,5) ]:} Verify that (A^(-1) )^(-1) =...

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  10. Evaluate {:[( x,y , x+y),( y,x+y,x),( x+y,x,y)]:}

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  11. Evaluate {:[(1,x,y),( 1,x+y,y),( 1,x,x+y)]:}

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  12. Using properties of determinants in Exercises prove that : {:[( alph...

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  13. {:[( x,x^(2) , 1+ px^(3) ),( y,y^(2) , 1+ py^(3)),( z,z^(2) , 1+pz^(3)...

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  14. solve {:[( 3a,-a+b,-a+c),( -b+a, 3b,-b+c) ,( -c+a,-c+b,3c) ]:} = 3( a+...

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  15. {:[( 1,1+p,1+p+q),( 2,3+2p,4+3p+2q),( 3,6+3p,10+6p+3q)]:}=1

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  16. Without expanding evaluate the determinant |(sinalpha,cosalpha,sin(alp...

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  17. Solve system of linear equations , using matrix method 2x+ 3y + ...

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  18. If a,b,c are in A.P. then the determinant {:[( x+2,x+3,x+2a),( x+3,...

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  19. If x,y,z are nonzero real number , then the inverse of matrix A= {:[( ...

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  20. Let A={:[( 1,sin theta , 1),( -sin theta , 1, sin theta ),( -1 ,-sin ...

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