The value of determinant #|(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)|# is?

1 Answer
maganbhai P.
Aug 1, 2018

#D=|(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)|=(a+b+c)^3#

Explanation:

Here .

#D=|(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)|#

#R_1+R_2+R_3#
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#color(red)(a-b-c+2b+2c),color(blue)(2a+b-c-a+2c),color(green)(2a+2b+c-a-b#
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#:.D=|(a+b+c,a+b+c,a+b+c),(2b,b-c-a,2b),(2c,2c,c-a-b)|#

#R_1(1/(a+b+c))#

#:.D=(a+b+c)|(1,1,1),(2b,b-c-a,2b),(2c,2c,c-a-b)|#

#C_2-C_1 and C_3-C_1#

#:.D=(a+b+c)|(1,1-1,1-1),(2b,b-c-a-2b,2b-2b),(2c,2c-2c,c-a-b-2c)|#

#:.D=(a+b+c)|(1,0,0),(2b,-b-c-a,0),(2c,0,-c-a-b)|#

Expansion of determinant:

#:.D=(a+b+c){1(-b-c-a)(-c-a-b)-0-0}#

#:.D=(a+b+c)(b+c+a)(c+a+b)#

#:.D=(a+b+c)^3#

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