<p>If \(A = \begin{bmatrix} 0 & x \\ y & 0 \end{bmatrix}\) and \(A^3 + A = O\), then sum of possible values of \(xy\) is</p>
Step-by-Step Solution
Key Concept: Calculate A³ by finding A² first, then use the condition A³ + A = O to extract a constraint on the product xy. The constraint will be a polynomial equation whose roots are the possible values of xy.
<p><strong>Step 1:</strong> Calculate A²</p><p>A² = $\begin{bmatrix} 0 & x \\ y & 0 \end{bmatrix}\begin{bmatrix} 0 & x \\ y & 0 \end{bmatrix} = \begin{bmatrix} xy & 0 \\ 0 & xy \end{bmatrix} = (xy)I$</p><p><strong>Step 2:</strong> Calculate A³</p><p>A³ = A · A² = $\begin{bmatrix} 0 & x \\ y & 0 \end{bmatrix}(xy)I = (xy)\begin{bmatrix} 0 & x \\ y & 0 \end{bmatrix} = (xy)A$</p><p><strong>Step 3:</strong> Apply the condition A³ + A = O</p><p>$(xy)A + A = O$</p><p>$A[(xy) + 1] = O$</p><p><strong>Step 4:</strong> Determine when this holds</p><p>Since A is not the zero matrix (it has non-zero entries x and y in general), we need:</p><p>$(xy) + 1 = 0$</p><p>$xy = -1$</p><p><strong>Step 5:</strong> Find sum of possible values</p><p>The only possible value of xy is -1.</p><p>∴ Sum of possible values = <strong>-1</strong></p><p>Answer: B</p>
Correct Answer: B