Which determinant does NOT equal vmatrix1&x&x2\\1&y&y2\\1&z&z2vmatrix?

$$\begin{vmatrix}1&x+1&x^2+1\\1&y+1&y^2+1\\1&z+1&z^2+1\end{vmatrix}$$

$$\begin{vmatrix}0&x-y&x^2-y^2\\0&y-z&y^2-z^2\\1&z&z^2\end{vmatrix}$$

$$\begin{vmatrix}2&x+y&x^2+y^2\\2&y+z&y^2+z^2\\1&z&z^2\end{vmatrix}$$

GATE 2013 CS (CS1) – Question 13

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Which determinant does NOT equal $$\begin{vmatrix}1&x&x^2\\1&y&y^2\\1&z&z^2\end{vmatrix}?$$

$$\begin{vmatrix}1&x(x+1)&x+1\\1&y(y+1)&y+1\\1&z(z+1)&z+1\end{vmatrix}$$
$$\begin{vmatrix}1&x+1&x^2+1\\1&y+1&y^2+1\\1&z+1&z^2+1\end{vmatrix}$$
$$\begin{vmatrix}0&x-y&x^2-y^2\\0&y-z&y^2-z^2\\1&z&z^2\end{vmatrix}$$
$$\begin{vmatrix}2&x+y&x^2+y^2\\2&y+z&y^2+z^2\\1&z&z^2\end{vmatrix}$$

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Correct answer: (A) $$\begin{vmatrix}1&x(x+1)&x+1\\1&y(y+1)&y+1\\1&z(z+1)&z+1\end{vmatrix}$$

Explanation

Write original columns C₁,C₂,C₃. A uses C₁,C₂+C₃,C₁+C₂; subtracting C₁ from its last column leaves C₂, so its determinant is det(C₁,C₃,C₂)=−D. B adds C₁ to other columns and preserves D. C replaces rows by R₁−R₂,R₂−R₃,R₃ through determinant-one transformations; D uses R₁+R₂,R₂+R₃,R₃, also determinant-one. Thus A is not identically equal.

GATE 2013 CS (CS1) – Question 13

Explanation

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