UPSC NDA I 2019 Mathematics Solved Paper

Concept:An involuntary matrix is a square matrix $A$ such that $A^2 = I$, where $I$ is the identity matrix of the same order.Explanation:Given $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.Compute $A^2$ by multiplying $A$ with itself:$A^2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.Perform matrix multiplication:First row, first column: $(0 \times 0) + (1 \times 1) = 0 + 1 = 1$.First row, second column: $(0 \times 1) + (1 \times 0) = 0 + 0 = 0$.Second row, first column: $(1 \times 0) + (0 \times 1) = 0 + 0 = 0$.Second row, second column: $(1 \times 1) + (0 \times 0) = 1 + 0 = 1$.Thus $A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$.Since $A^2 = I$, $A$ satisfies the condition for an involuntary matrix.It is not singular because $|A| = -1 \neq 0$, not nilpotent because $A^2 \neq O$, and not idempotent because $A^2 \neq A$.Answer: Involuntary matrix.