<p>The determinant \(\begin{vmatrix} y^2-xy & x^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix}\) is equal to \(\begin{vmatrix} bx-ay & cx-by \\ a'x-b'y & b'x-c'y \end{vmatrix}\)</p>
Step-by-Step Solution
Key Concept: The left side is a 3×3 determinant with the first row containing two elements (incomplete), while the right side is a 2×2 determinant. These are fundamentally different matrix dimensions and cannot be equal as stated.
<p><strong>Step 1:</strong> Analyze the left side determinant structure. The notation shows a 3×3 determinant with rows: [y²-xy, x²], [a, b, c], [a', b', c']. The first row has only 2 elements instead of 3, making this notation incomplete/invalid as written.</p><p><strong>Step 2:</strong> Analyze the right side. This is clearly a 2×2 determinant with defined rows: [bx-ay, cx-by] and [a'x-b'y, b'x-c'y].</p><p><strong>Step 3:</strong> Compare dimensions. A 3×3 determinant (even if somehow defined) produces a scalar from a 3×3 matrix, while a 2×2 determinant produces a scalar from a 2×2 matrix. While both produce scalars, the dimensional inconsistency of the left side matrix and the algebraic impossibility of equating these for arbitrary variables shows the statement cannot be universally true.</p><p><strong>Step 4:</strong> Conclude. The statement as written contains a structural error (incomplete 3×3 matrix on the left). The equality does not hold as a mathematical identity. For arbitrary values of variables and constants, these expressions are not generally equal.</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B