Question

Find the determinant of the matrix A=\(\begin{bmatrix}1&x&y\\1&x&-y\\1&-x^2&y^2\end{bmatrix}\).

(a) (x+1)

(b) -2xy(x+1)

(c) xy(x+1)

(d) 2xy(x+1)

I had been asked this question in final exam.

Question is taken from Properties of Determinants topic in portion Determinants of Mathematics – Class 12

Answer 1

Right option is (b) -2xy(x+1)

Best explanation: Given that, A=\(\begin{bmatrix}1&x&y\\1&x&-y\\1&-x^2&y^2\end{bmatrix}\)

Δ=\(\begin{vmatrix}1&x&y\\1&x&-y\\1 &-x^2&y^2 \end{vmatrix}\)

Taking x common C2 and y common from C3, we get

Δ=xy\(\begin{vmatrix}1&1&1\\1&1&-1\\1&-x&y\end{vmatrix}\)

Expanding along R1, we get

Δ=xy{1(y-x)-1(y+1)+1(-x-1)}

Δ=xy(y-x-y-1-x-1)

Δ=xy(-2x-2)=-2xy(x+1).