|{:(1+a,1,1),(1,1+a,1),(1,1,1+a):}|=a^3+...

`|{:(1+a,1,1),(1,1+a,1),(1,1,1+a):}|=a^3+3a^2`

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if A={:[(1,1,1),(1,1,1),(1,1,1)]:}, prove by mathematical induction that, A^(n)={:[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))]:} for every positive integer n.

If A= {:[( 2,-1,1),(-1,2,-1),(1,-1,2) ]:} Verify that A^(3) -6A^(2) +9A -4I=O and hence find A^(-1)

Prove that , |{:(1+a_1," "1," "1),(" "1,1+a_2," "1),(" "1," "1,1+a_3):}|=a_1a_2a_3(1+(1)/(a_1)+(1)/(a_2)+(1)/(a_3))

Find A^(-1) if A=|(0,1,1),(1,0,1),(1,1,0)| and show that A^(-1)=(A^(2)-3I)/2

A={:[( 1,1,1),(1,2,-3),(2,-1,3)]:} Show that A^(3) - 6A^(2) +5A +11 I =O. Hence , find A^(-1)

Let A =[(1,-1,1),(2,1,-3),(1,1,1)] and 10B=[(4,2,2),(-5,0,alpha),(1,-2,3)] . If B is the inverse of A, then find the value of alpha

If A={:[(0,1,2),(1,2,3),(3,1,1)]and B={:[(2,1,3),(-1,0,1),(3,-1,4)], show that, AB ne BA .

Let omega=-1/2+i(sqrt(3))/2 . Then the value of the determinant |(1,1,1),(1,-1-omega^2,omega^2),(1,omega^2,omega^4)| is (A) 3omega (B) 3omega(omega-1) (C) 3omega^2 (D) 3omega(1-omega)

If A=1/3({:(-1,2,-2),(-2,1,2),(2,2,1):}) , show that "AA"^(T)=I_(3) .

Show that (i) [(5,-1),(6,7)][(2,1),(3,4)] ne [(2,1),(3,4)][(5,-1),(6,7)] (ii) [(1,2,3),(0,1,0),(1,1,0)][(-1,1,0),(0,-1,1),(2,3,4)] ne [(-1,1,0),(0,-1,1),(2,3,4)][(1,2,3),(0,1,0),(1,1,0)]

CHHAYA PUBLICATION-DETERMINANT -EXERCISE 2B

|{:(1+x,1,1),(1,1+y,1),(1,1,1+z):}|=xy+yz+zx+xyz
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|{:(x+4,2x,2x),(2x,x+4,2x),(2x,2x,x+4):}|=(5x+4)(x-4)^2
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|{:(1+a,1,1),(1,1+a,1),(1,1,1+a):}|=a^3+3a^2
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|{:(1,b+c,b^2+c^2),(1,c+a,c^2+a^2),(1,a+b,a^2+b^2):}|=(b-c)(c-a)(a-b)
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|{:(a^2,ab,ac),(ab,b^2,bc),(ca,bc,c^2):}|=?
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|{:(-1,b,c),(a,-1,c),(a,b,-1):}|=(a+1)(b+1)(c+1)((a)/(a+1)+(b)/(b+1)+(...
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|{:(x^2+y^2+1,x^2+2y^2+3,x^2+3y^2+4),(y^2+2,2y^2+6,3y^2+8),(y^2+1,2y^2...
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|{:(" "3a," "-a+b,-a+c),(a-b," "3b," "c-b),(a-c," "b-c," "3c):}|...
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|{:(x,a,b),(a,x,b),(a,b,x):}|=(x-a)(x-b)(x+a+b)
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|{:(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c):}|=3abc-a^3-b^3-c^3
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Prove |{:(1,a^2+bc,a^3),(1,b^2+ca,b^3),(1,c^2+ab,c^3):}|=-(a-b)(b-c)(c...
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|{:(a^2,a^2-(b-c)^2,bc),(b^2,b^2-(c-a)^2,ca),(c^2,c^2-(a-b)^2,ab):}|=(...
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|{:((a+1)(a+2),a+2,1),((a+2)(a+3),a+3,1),((a+3)(a+4),a+4,1):}|=-2
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Show that the following determinant is a perfact square : |{:(1,a,a^...
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if x+y+z = 0 then prove that |{:(x,y,z),(x^2,y^2,z^2),(y+z,z+x,x+y):}|...
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|{:(1,1,1),(x,y,z),(x^3,y^3,z^3):}|=0
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|{:(x," "4,-2),(4," "x,-2),(4,-2," "x):}|=0
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|{:(x,a,a),(a,x,b),(b,b,x):}|=0
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|{:(x+1," "2," "3),(" "1,x+1," "3),(" "3,-6,x+1):}|=0
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|{:(a-x," "b," "c),(" "b,c-x," "a),(" "c," "a,b-x):}|=0
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