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Prove that : |{:(a,b,c),(b,c,a),(c,a,b):...

Prove that : `|{:(a,b,c),(b,c,a),(c,a,b):}|=3 a b c-a^(3)-b^(3)-c^(3)`

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To prove the determinant \[ \left| \begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array} \right| = 3abc - a^3 - b^3 - c^3, ...
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Prove: \ |(a, b, c),( a-b,b-c,c-a),( b+c,c+a, a+b)|=a^3+b^3+c^3-3a b c

Prove: |[b+c ,a-b ,a], [c+a, b-c ,b], [a+b, c-a, c]|=3a b c-a^3-b^3-c^3

Prove that |[b+c ,a-b, a] ,[c+a, b-c ,b],[ a+b, c-a, c]|=3abc-a^3-b^3-c^3dot

Prove that |{:(a,,a^(2),,bc),(b ,,b^(2),,ac),( c,,c^(2),,ab):}| = |{:(1,,1,,1),(a^(2) ,,b^(2),,c^(2)),( a^(3),, b^(3),,c^(3)):}|

Prove that : (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3a b c)

Prove that : (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3a b c)

Prove: |(0,b^2a, c^2a),( a^2b,0,c^2b),( a^2c, b^2c,0)|=2a^3b^3c^3

Using properties of determinants, prove that: |{:(a, a +b, a+b+c),(2a, 3a + 2b, 4a + 3b + 2c),(3a, 6a+3b, 10a + 6b + 3c):}| = a^(3)

NAGEEN PRAKASHAN ENGLISH-DETERMINANTS-Miscellaneous Exercise
  1. Prove that : |{:(a,b,c),(b,c,a),(c,a,b):}|=3 a b c-a^(3)-b^(3)-c^(3)

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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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