Prove: |[1,a^2+bc, a^3],[ 1,b^2+c a, b^3...

Prove: `|[1,a^2+bc, a^3],[ 1,b^2+c a, b^3],[ 1,c^2+a b, c^3]|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)`

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Prove: |a^3 2a b^3 2b c^3 2c|=2(a-b)(b-c)(c-a(a+b+c)

By using properties of determinants. Show that: (i) |[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|=(a-b)(b-c)(c-a) (ii) |[1, 1, 1],[a, b, c],[ a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a+b+c)

Prove: |(1,b+c ,b^2+c^2),( 1,c+a ,c^2+a^2),( 1,a+b ,a^2+b^2)|=(a-b)(b-c)(c-a)

Show that |[a ,b ,c],[ a^2,b^2,c^2],[bc, ca, ab]|=|[1, 1, 1],[a^2,b^2,c^2],[a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a b+b c+c a) .

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

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

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

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Prove: |((a+1)(a+2),(a+2),1 ),((a+2)(a+3),(a+3),1), ((a+3)(a+4) ,(a+4)...
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Prove: |(a^2,a^2-(b-c)^2,b c), (b^2,b^2-(c-a)^2,c a),( c^2,c^2-(a-b)^2...
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Prove: |[1,a^2+bc, a^3],[ 1,b^2+c a, b^3],[ 1,c^2+a b, c^3]|=-(a-b)(b-...
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Prove: |(a^2,b c, a c+c^2),(a^2+a b,b^2,a c ),(a b,b^2+b c,c^2)|=4a^2b...
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Prove: |(x+4,x,x),(x,x+4,x),(x,x,x+4)|=16(3x+4)
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Show that |[1,1+p,1+p+q],[2,3+2p,1+3p+2q],[3,6+3p,1+6p+3q]|=1
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Prove: |(a, b-c,c-b),( a-c, b, c-a),( a-b,b-a, c)|=(a+b-c)(b+c-a)(c+a-...
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Prove: |[a^2, 2ab,b^2],[b^2,a^2, 2ab],[2ab,b^2,a^2]|=(a^3+b^3)^2
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Prove: |(a^2+1,a b ,a c), (a b,b^2+1,b c), (c a,cb,c^2+1)|=1+a^2+b^2...
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Prove: |(1,a, a^2),(a^2, 1,a),( a, a^2, 1)|=(a^3-1)^2
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Prove: |a+b+c-c-b-c a+b+c-a-b-a a+b+c|=2(a+b)(b+c)(c+a)
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Prove: |[b+c,a,a],[b,c+a,b],[c,c,a+b]|=4a b c
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Show that |[b^2+c^2,ab,ac],[ba,c^2+a^2,bc],[ca,cb,a^2+b^2]|=4a^2b^2c^2
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Prove: |(0,b^2a, c^2a),( a^2b,0,c^2b),( a^2c, b^2c,0)|=2a^3b^3c^3
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Prove: |((a^2+b^2)/c,c,c),( a,(b^2+c^2)/a ,a),( b,b,(c^2+a^2)/b)|=4a b...
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Prove: |(-bc, b^2+b c,c^2+bc), (a^2+a c,-a c,c^2+a c),( a^2+a b,b^2+a ...
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By using properties of determinants. Show that: (i) |x+4 2x2x2xx+4 2x2...
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show that |{:(y+k,y,y),(y,y+k,y),(y,y,y+k):}|=k^(2)(3y+k)
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Prove: |(y+z, z, y),( z, z+x,x),( y, x,x+y)|=4\ x y z
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Show that | (-a(b^2 + c^2 - a^2), 2b^3, 2c^3), (2a^3, -b(c^2 + a...
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