prove, using the properties of determinants:

  • determinant | a+b+c -c -b -c a+b+c -a -b -a a+b+c| = 2(a+b)(b+c)(c+a)

Let A= a+b+c-c-b-ca+b+c-a-b-aa+b+cusing C1C1+C2 and C2C3+C2 =a+b-c-b-ba+bb+c-a-b-ab+ca+b+c=a+bb+c1-1-b11-a-11a+b+cusing C1C1+C2 and C2C3+C2 =a+bb+c0-1-b21-a01a+b+cNow expanding along C1, =a+bb+c-2-a-b-c+b=2a+bb+ca+c

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