Question

Find the determinant of A = \(\begin{bmatrix} x^2 & y^2 & -2xy\\ x & y &0\\1&1&2\end{bmatrix}\), if x + y = 0
5. None of these

Answer 1

Correct Answer - Option 1 : 0

Concept:

Properties of determinants:

• If each entry in any row or column of a determinant is 0, then the value of the determinant is zero.
• If we interchange any two rows (columns) of a matrix, then the determinant is multiplied by -1.
• If any two rows (columns) of a matrix are same then the value of the determinant is zero.

Calculation:

A = \(\begin{bmatrix} x^2 & y^2 & -2xy\\ x & y &0\\1&1&2\end{bmatrix}\) 

|A| = \(\begin{vmatrix} x^2 & y^2 & -2xy\\ x & y &0\\1&1&2\end{vmatrix}\)

c₃ = c₃ - c₁ - c₂

|A| = \(\begin{vmatrix} x^2 & y^2 & -(2xy+x^2+y^2)\\ x & y &0-x-y\\1&1&2-1-1\end{vmatrix}\)

|A| = \(\begin{vmatrix} x^2 & y^2 & -(x+y)^2\\ x & y &-(x+y)\\1&1&0\end{vmatrix}\)

|A| = \(\begin{vmatrix} x^2 & y^2 & 0\\ x & y &0\\1&1&0\end{vmatrix}\)

|A| = 0