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Using properties of determinants, prove that if `x, y, z` are different and `Delta = abs{:(x,x^2, 1+ x^3),(y , y^2, 1+ y^3),(z,z^2, 1+ z^3):} = 0`, then `1+ xyz = 0`.

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MODERN PUBLICATION-DETERMINANTS-EXERCISE
  1. Let a, b, and c be three real numbers satisfying [(a,b,c)][[1,9,7],[...

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  2. Let omega be a solution of z^3-1 = 0 If a = 2 , b=8 and c =7 then the ...

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  3. If A= ((1,0,0),(2,1,0),(3,2,1)), U(1), U(2), and U(3) are column matri...

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  4. A complex number U = 4+2i.The value of |U| is

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  5. Let A be the set of all 3× 3 symmetric matrices all of whose either 0 ...

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  6. Let A be the set of all 3xx3 symmetric matrices all of whoes entries...

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  7. For a 2xx2 matrix, A = [a(ij)], whose elements are given by a(ij) = ...

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  8. If A is a square matrix such that A^2 = A, then write the value of (I+...

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  9. Evaluate the following determinants: {:|(3,x),(x,1)| = |(3,2),(4,1)|

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  10. If Delta={:|(1,2,3),(2,0,1),(5,3,8)|, write the minor of the element a...

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  11. If A = [[3,1],[-1,2]], show thatA^2-5A +7I = O

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  12. If A = [[costheta,sintheta],[-sintheta,costheta]] then prove that A^n...

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  13. If A = {:[(cosalpha, sinalpha),(-sinalpha, cosalpha)], A'A=I.

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  14. Using elementary transformations find the inverse of [[2,0,-1],[5,1,0]...

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  15. Using properties of determinants, prove that if x, y, z are different ...

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  16. If A = {:[(1,2,3),(0,-1,4),(3,2,1)], find (A')^-1

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  17. Solve by matrix method 2/x + 3/y + 10/z = 4 , 4/x - 6/y + 5/z = 1 , ...

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  18. Let A = [[0,1],[0,0]] , show that (aI + bA)^n = a^nI + na^(n-1) bA, w...

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  19. Using properties of determinants, show that: |[[(b+c)^2, a^2, a^2],[b^...

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  20. Use product [[1,-1,2],[0,2,-3],[3,-2,4]][[-2,0,1],[9,2,-3],[6,1,-2]] t...

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