Q46 of 256 Page 7

Without expanding, prove that .

Let

We know that the sign of a determinant changes if any two rows or columns are interchanged.

By interchanging R₁ and R₂, we get

By interchanging R₂ and R₃, we get

Hence,

Let us once again consider

By interchanging R₁ and R₂, we get

By interchanging C₁ and C₂, we get

Recall that the value of a determinant remains same if it its rows and columns are interchanged.

Hence,

Thus,

Prove the following identities –

= 2(a - b)(b - c)(c - a)(a + b + c)

Show that where a,b,c are in A.P.

Show that where α, β and γ are in A.P.