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[,a+b,b+c,c+a],[2a,0,b+c,c+a,a+b],[,c+d,a+b,b+c]

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Prove that: |[a+b, b+c, c+a],[b+c,c+a,a+b],[c+a,a+b,b+c]|=2|[a,b,c],[b,c,a],[c,a,b]|

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is (a) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is (a) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

If D=|[a,b,c], [c,a,b], [b,c,a]| and D'=|[b+c, c+a, a+b], [a+b, b+c, c+a], [c+a, a+b, b+c]|, then prove that D' = 2D

Let D_1=|[a, b, a+b], [c, d, c+d], [a, b, a-b]| and D_2=|[a, c, a+c], [b, d, b+d], [a, c, a+b+c]| then the value of |(D_1)/(D_2)| , where b!=0 and a d!=b c , is _____.

Let D_1=|[a, b, a+b], [c, d, c+d], [a, b, a-b]| and D_2=|[a, c, a+c], [b, d, b+d], [a, c, a+b+c]| then the value of |(D_1)/(D_2)| , where b!=0 and a d!=b c , is _____.

Prove that |[a+b+c, -c, -b],[-c, a+b+c, -a],[-b, -a, a+b+c]|=2(a+b)(b+c)(c+a)

If |(b +c,c +a,a +b),(a +b,b +c,c +a),(c +a,a +b,b +c)| = k |(a,b,c),(c,a,b),(b,c,a)| , then the value of k, is 1 b. 2 c. 3 d. 4

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