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Prove that |{:(1,x+alpha,y+z-alpha),(...

Prove that
`|{:(1,x+alpha,y+z-alpha),(1,y+beta, z+x-beta),(1,z+gamma,x+y-gamma):}|=0`

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`L.H.S.=|{:(1,x+alpha,y+z-alpha),(1,t+beta,+x-beta),(1,z+gamma,x+y-gamma):}|`
`=|{:(1,x+alpha,y+z-alpha),(1,t+beta,+x-beta),(1,z+gamma,x+y-gamma):}|(C_(3)toC_(3)+C_(2))`
`=(x+y+z)|{:(1,x+alpha,1),(1,t+beta,1),(1,z+gamma,1):}|`
`=0 " "(therefore C_(1)" and "C_(3)" are same")`
`=R.H.S." "Hence Proved`.
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NAGEEN PRAKASHAN ENGLISH-DETERMINANTS-Miscellaneous Exercise
  1. Prove that |{:(1,x+alpha,y+z-alpha),(1,y+beta, z+x-beta),(1,z+gamma...

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  2. Prove that the determinant [xsinthetacostheta-sintheta-x1costheta1x]is...

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  3. Without expanding the determinant, prove that |a a^2b c bb^2c a cc^2a ...

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  4. Ecaluate [{:(cosalphacosbeta,cosalphasinbeta,-sinalpha),(-sinbeta,co...

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  5. " if a,b, and c are real number and " |{:(b+c,,c+a,,a+b),(c+a,,a+b,...

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  6. Solve the following determinant equation: |[x+a, b, c], [c, x+b, a],...

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  7. Using properties of determinants, prove that |[a^2, bc, ac+c^2] , [a^...

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  8. If A^(-1)=[3-1 1-15 6-5 5-2 2] and B=[1 2-2-1 3 0 0-2 1] , find (A B)^...

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

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  10. Evaluate |x y x+y y x+y xx+y x y| .

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  11. Evaluate the following: |[1,x,y],[1, x+y, y],[1, x, x+y]|

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  12. Using properties of determinants. Prove that|alphaalpha^2beta+gammarho...

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  13. For any scalar p prove that =|[x,x^2, 1+p x^3],[y, y^2, 1+p y^3],[z, z...

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  14. show that |{:(3a,,-a+b,,-a+c),(-b+a ,,3b,,-b+c),( -c+a,, -c+b,,3c):}| ...

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  15. Show that |(1, 1+p, 1+p+q), (2, 3+2p, 4+3p+2q), (3, 6+3p, 10+6p+3q)|=1...

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  16. Show that |[sinalpha, cosalpha, cos(alpha+delta)],[sinbeta, cosbeta, ...

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  17. Solve the system of equations2/x+3/y+(10)/z=4,4/x-6/y+5/z=1,6/x+9/y-(2...

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  18. Choose the correct answer in questions 17 to 19: If a, b, c are in ...

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  19. If x, y, z are non-zero real numbers, then the inverse of matrix A=[(...

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  20. Let A=|1sintheta1-sintheta1sintheta-1-sintheta1|, where 0lt=thetalt=2p...

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