Write the value of the determinant |(a, 1, b+c)(b, 1, c+a)(c, a, a+b)|.

Question

Write the value of the determinant \(\begin{vmatrix} a & 1 & b+c \\ b & 1 & c+a \\ c & a & a+b \end{vmatrix}.\)

Answer 1

Using the property that if the equimultiples of corresponding elements of other rows (or columns) are added to every element of any row (or column) of a determinant, then the value of determinant remains the same.

Using column transformation, C₁→C₁+C₃

Using the property that if each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

Taking out factor(a + b + c) from C₁,

We get,

Using column transformation, C₁→C₁-C₂

We get,

Expanding along C₁, we get

∆ = (a + b + c) × [(1 - a)(c + a - (b + c))]=(1 - a)(a - b)(a + b + c)