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If A = [((1)/(2),alpha),(0,(1)/(2))] , p...

If A = `[((1)/(2),alpha),(0,(1)/(2))]` , prove that `sum_(k=1)^(n)"det" (A^(k))=(1)/(3)(1-(1)/(4^(n))).`

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SURA PUBLICATION-MATRICES AND DETERMINANTS -EXERCISE 7.2
  1. Prove that |(a^(2),bc,ac+c^(2)),(a^(2)+ab,b^(2),ac),(ab,b^(2)+bc,c^(2)...

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  2. Prove that |(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc(1+(1)/(a)+(1)/(b)+(1)...

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  3. Prove that |(sec^(2)theta,tan^(2)theta,1),(tan^(2)theta,sec^(2)theta,-...

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  4. Show that |(x+2a,y+2b,z+2c),(x,y,z),(a,b,c)|=0.

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  5. Write the general form of a 3xx3 skew -symmetric matrix and prove that...

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  6. If |(a,b,aalpha+b),(b,c,balpha+c),(aalpha+b,balpha+c,0)|=0, prove that...

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  7. Prove that |(1,a,a^(2)-bc),(1,b,b^(2)-ca),(1,c,c^(2)-ab)|= 0.

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  8. If a,b,c are p^(pt),q^(th) and r^(th) terms of an A.P., find the value...

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  9. Show that |(a^(2)+x^(2),ab,ac),(ab,b^(2)+x^(2),bc),(ac,bc,c^(2)+x^(2))...

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  10. If a,b,c are all positive, and are p^(th), q^(th) and r^(th) terms of ...

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  11. Find the value of |(1,log(x) y,log(x) z),(log(y) x,1,log(y) z),(log(z)...

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  12. If A = [((1)/(2),alpha),(0,(1)/(2))] , prove that sum(k=1)^(n)"det" (A...

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  13. Without expanding, evaluate the determinants : |(2,3,4),(5,6,8),(6x...

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  14. Without expanding evaluate the folleing determinents (ii)|(x+y,y+z,z+x...

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  15. If A is a square matrix and |A|=2, find the value of |"AA"|^(T) .

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  16. If A and B are square matrices of order 3 such that |A| = -1 and |B| =...

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  17. If lambda = -2, determine the value of |(0,2lambda,1),(lambda^(2),0,3l...

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  18. Determine the roots of the equation |(1,4,20),(1,-2,5),(1,2x,5x^(2))|=...

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  19. Verify that det (AB) = (det A) (det B) for A=[(4,3,-2),(1,0,7),(2,3...

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  20. Using cofactors of elements of second row, evaluate |A|, where A = [(5...

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