Q. 164.0( 4 Votes )

# Using properties of determinants prove that:

Answer :

[R_{1}’ = R_{1} + R_{2} + R_{3}]

[R_{1}’ = R_{1}/(a + b + c)]

= (a + b + c)[2(b - c)c - b(c - a) + (c + a)(c - a) - (a + b)(b - c)] [expansion by first row]

= (a + b + c)(2bc - 2c^{2} - bc + ab + c^{2} - a^{2} - ab - b^{2} + ac + bc

= (a + b + c)(ab + bc + ac - a^{2} - b^{2} - c^{2})

= **3abc - a ^{3} - b^{3} - c^{3}**

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