Prove, using properties of determinant : |(y+k,y,y)(y,y+k,y)(y,y,y+k)|=k^2(3y+k).

Question

Prove, using properties of determinant :

\(\begin{vmatrix} y+k & y & y \\[0.3em] y & y+k & y \\[0.3em] y & y &y+k \end{vmatrix}=k^2(3y+k)\)

Answer 1

LHS,

\(\begin{vmatrix} y+k & y & y \\[0.3em] y & y+k & y \\[0.3em] y & y &y+k \end{vmatrix}\) 

Expanding along C₁ we get

= (3y + k) {1(k² - 0) - 0 + 0}

= (3y + k) . k²

= k²(3y + k)