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Prove that |{:(a,a+b,a=b+c),(2a,3a+2b,4a...

Prove that `|{:(a,a+b,a=b+c),(2a,3a+2b,4a+3b+2c),(3a,6a+3b,10a+6b+3c):}|=a^3`

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Using properties of determinants in Exercise 11 to 15 prove that |{:(3a,-a+b,-a+c),(-b+a,3b,-b+c),(-c+a,-c+b,3c):}|=3(a+b+c)(ab+bc+ca)

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Without expanding the determinant prove that |{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}|=|{:(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3):}|

Prove that: |(-2a , a+b , a+c ) ( b+a ,-2b , b+c ) ( c+a, c+b , -2c)| =4(a+b)(b+c)(c+a)

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b):}|=(a+b+c)^3

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(1,1,1),(a,b,c),(a^3,b^3,c^3):}|=(a-b)(b-c)(c-a)(a+b+c)

Calculate the Miller indices of crystal planes which cut through the crystal axes at (i) (2a, 3b, c) (ii) (a, b, c) (iii) (6a, 3b, 3c) and (iv) (2a, -3b, -3c).

Prove that |b c-a^2c a-b^2a b-c^2-b c+c a+a bb c-c a+a bb c+c a-a b(a+b)(a+c)(b+c)(b+a)(c+a)(c+b)|=3.(b-c)(c-a)(a-b)(a+b+c)(a b+b c+c a)

Using the properties of determinants, prove the following |{:((b+c)^2,a^2,a^2),(b^2,(c+a)^2,b^2),(c^2,c^2,(a+b)^2):}|=2abc(a+b+c)^3

bar(a)=a_(1)hati+a_(2)hatj+a_(3)hatk,bar(b)=b_(1)hati+b_(2)hatj+b_(3)hatk,bar( c )=c_(1)hati+c_(2)hatj+c_(3)hatk are three non zero vectors. The unit vector bar( c ) is perpendicular to bar(a) and bar(b) . The angle between bar(a) and bar(b) is (pi)/(6) then, |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}| = .............

KUMAR PRAKASHAN-DETERMINANTS -Textbook Illustrations for Practice work
  1. Evaluate {:[( 102,18,36),( 1,3,4),( 17,3,6)]:}

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  2. Using the property of determinants and without expanding prove the fol...

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  3. Prove that |{:(a,a+b,a=b+c),(2a,3a+2b,4a+3b+2c),(3a,6a+3b,10a+6b+3c):}...

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  4. Without expanding prove that Delta=|{:(x+y,y+z,z+x),(z,x,y),(1,1,1):...

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  5. Evaluate Delta=|{:(1,a,bc),(1,b,ca),(1,c,ab):}|

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  6. Prove that {:[( b+c,a,a), ( b,c+a,b),( c,c,b+a) ]:}

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  7. If x,y,z are different and Delta=|{:(x,x^2,1+x^3),(y,y^2,1+y^3),(z,z...

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

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  9. Find the area of the triangle whose vertices are (3,8),(-4,2) and (5,1...

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  10. Find the equation of the line joining A(1,3) and B(0,0) using determin...

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  11. Find the minor of elements 6 in the determinant Delta=|{:(1,2,3),(4,...

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  12. Find minors and cofactors of all the elements of the determinant |{:(1...

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  13. Find minors and cofactors of the elements a(11),a(21) in the determina...

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  14. Find minors and cofactors of the elements of the determinant |{:(2,-3,...

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  15. Find adj A for A = {:[( 2,3),( 1,4) ]:}

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  16. If A=[{:(1,3,3),(1,4,3),(1,3,4):}] , then verify that A adjA=|A|I. Als...

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  17. If A=[{:(2,3),(1,-4):}] and B=[{:(1,-2),(-1,3):}] then verify that (AB...

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  18. Show that the matrix A=[{:(2,3),(1,2):}] satisfies the equation A^2-4A...

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  19. Solve the system of equations {:(2x+5y=1),(3x+2y=7):},

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  20. Solve system of linear equations , using matrix method if exists 4x-...

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