prove, using the properties of determinants: determinant |-a(b^2+c^2-a^2) 2b^3 2c^3 2a^3 -b(c^2+a^2-b^2) 2c^3 2a^3 2b^3 -c(a^2+b^2-c^2)| = abc(a^2+b^2+c^2)^3 - Maths - Determinants

Question

prove, using the properties of determinants:

determinant |-a(b^2+c^2-a^2) 2b^3 2c^3 2a^3 -b(c^2+a^2-b^2)
2c^3 2a^3 2b^3 -c(a^2+b^2-c^2)| = abc(a^2+b^2+c^2)^3

Anuradha Sharma · Accepted Answer

Given , A= -ab2+c2-a22b32c32a3-bc2+a2-b22c32a32b3-ca2+b2-c2=abc -b2+c2-a22b22c22a2-c2+a2-b22c22a22b2-a2+b2-c2Now performing C1→C1+C2+C3=abc b2+c2+a22b22c2b2+c2+a2-c2+a2-b22c2b2+c2+a22b2-a2+b2-c2=abc b2+c2+a212b22c21-c2+a2-b22c212b2-a2+b2-c2Now performing R1→R1-R2 and R3→R3-R2=abc b2+c2+a20b2+c2+a201-c2+a2-b22c20b2+c2+a2-b2+c2+a2=abc b2+c2+a230101-c2+a2-b22c201-1Now expanding along R1, =abc b2+c2+a23-1×-1=abc b2+c2+a23