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Without expanding prove that: |[12, 2, 4...

Without expanding prove that: `|[12, 2, 4, -5, 1],[-8, 1, -5, 2, -1],[6, 4, a ,-3, 2],[-10, 2, 1 ,3 ,4],[-2, 4, 6 ,8, -5]|=|[12, -4, -8, 10, -2],[4, 1, -5, 2, -1],[-3, 4, a, -3, 2],[5, 2, 1, 3, 4],[1, 4, 6, 8, -5]|`.

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SHARAM PUBLICATION-DETERMINANT-EXAMPLE
  1. Without expanding find the value of the deteminant |[3, 6, 9],[-2, 4 ,...

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  2. Prove that:|[1, a ,a^2],[a^2, 1 ,a],[a, a^2, 1]| is a perfect square.

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  3. Without expanding prove that: |[12, 2, 4, -5, 1],[-8, 1, -5, 2, -1],[6...

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  4. Prove that: |[1, 1, 1],[a, b, c],[a^2, b^2, c^2]|=(a-b)(b-c)(c-a)

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  5. Solve : |[x-a, 0, 0],[a, x-b, 0],[a, b, x-c]|=0

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  6. What is the value of the determinant |[0, 0, 2],[0, 2, 0],[2, 0 ,0]|

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  7. Solve the following : [[1+x,1,1],[1,1+x,1],[1,1,1+x]]=0

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  8. Find the value of |{:(17,58,97),(19,60,99),(18,59,98):}| without expan...

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  9. Prove that the following. [[1+a,1,1],[1,1+b,1],[1,1,1+c]] = abc(1+1/...

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  10. Evaluate the following determinants: [[0,a^2,b],[b^2,0,a^2],[a,b^2,0...

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  11. Prove that: |[a+b+2c, a, b],[c, b+c+2a, b],[c, a, c+a+2b]|=2(a+b+c)^3

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  12. Without expanding prove that |[1,a,a^(2),-bc],[1,b,b^(2),-ca],[1,c,c^(...

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  13. Prove the following : [[1,bc,a(b+c)],[1,ca,b(c+a)],[1,ab,c(a+b)]]=0

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  14. Given the equations x=cy+bz, y=az+cx and z=bx+ay where x,y and z a...

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  15. Eliminate x,y,z from a=x/y-z, b=y/z-x, c=z/x-y

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  16. Prove that abs[[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]]=2abs[[a,b,c]...

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  17. Prove that the following. [[1+a,1,1],[1,1+b,1],[1,1,1+c]] = abc(1+1/...

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  18. Prove that |[x+y, x, x],[5x+4y, 4x, 2x], [10x+8y, 8x, 3x]|=x^3

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  19. Using properties of determinants prove that |{:(a+x,y,z),(x,a+y,z),...

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  20. Prove that the following. [[b+c,a,a],[b,c+a,b],[c,c,a+b]]=4ab

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