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Show that [[a1,b1,-c1],[-a2,b2,c2],[a3...

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`[[a_1,b_1,-c_1],[-a_2,b_2,c_2],[a_3,b_3,-c_3]]= [[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]`

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The system of equations a_1x+b_1y+c_1z=0 a_2x+b_2y+c_2z=0 a_3x+b_3y+c_3z=0 [[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]=0

Write the number of solution of the following system of equation. a_1x+b_1y+c_1z=0 a_2x+b_2y+c_2z=0 a_3x+b_3y+c_3z=0 and [[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]] =0

Show that by eliminating alpha and beta from the equations. a_ialpha+b beta_i+c_i =0, i=1,2,3 we get [[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]] =0

Write the value of k, if |{:(aa_1,aa_2,aa_3),(ab_1,ab_2,ab_3),(ac_1,ac_2,ac_3):}|=k|{:(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3):}|

Show that [[3a,-a+b,-a+c],[-b+a,3b,-b+c],[-c+a,-c+b,3c]] = 3(a+b+c)(ab+bc+ca)

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

Prove that [[1,a,a^2 - bc],[1,b,b^2 - ca],[1,c,c^2 - ab]]

ARIHANT PUBLICATION-DETERMINANTS -ODISHA BUREAU.S TEXTBOOK SOLUTIONS (EXERCISE 5(A))
  1. Show that x=1 is a solution of [[x+1,3,5],[2,x+2,5],[2,3,x+4]]=0

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  2. Show that (a+1) is a factor of abs([(a+1),2,3],[1,a+1,3],[3,-6,a+1])

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  3. Show that [[a1,b1,-c1],[-a2,b2,c2],[a3,b3,-c3]]= [[a1,b1,c1],[a2,b2,...

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  4. Prove that the following. [[a,b,c],[x,y,z],[p,q,r]]=[[y,b,q],[x,a,p],[...

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  5. Prove that |{:(1+a,1,1),(1,1+b,1),(1,1,1+c):}|=abc(1+1/a+1/b+1/c)or(ab...

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  6. Prove that the following. [[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]]=...

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  7. Prove that the following. [[(a+1)(a+2),a+2,1],[(a+2)(a+3),a+3,1],[(a+3...

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  8. Prove that the following. [[a+d,a+d+k,a+d+c],[c,c+b,c],[d,d+k,d+c]]=ab...

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  9. Prove that the following. [[1,1,1],[b+c,c+a,c+a],[b^2+c^2,c^2+a^2,a^2+...

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  10. Show that: abs((a,a^2,a^3),(b,b^2,b^3),(c,c^2,c^3))=abc(a-b)(b-c)(c-a)

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

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  12. Prove that the following. [[b^2+c^2,ab,ac],[ab,c^2+a^2,bc],[ca,cb,a^2+...

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  13. Prove that the following. [[a,b,c],[a^2,b^2,c^2],[bc,ca,ab]]=(b-c)(c-a...

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  14. Prove that the following. [[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]]=...

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  15. Prove that the following. |[(v+w)^2,u^2,u^2],[v^2,(w+u)^2,v^2],[w^2,w...

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  16. Factorize the following. [[x+a,b,c],[b,x+c,a],[c,a,x+b]]

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  17. Factorize the following. [[a,b,c],[b+c,c+a,a+b],[a^2,b^2,c^2]]

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  18. Factorize the following. [[x,2,3],[1,x+1,3],[1,4,x]]

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  19. Show that by eliminating alpha and beta from the equations. aialpha+...

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

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