The matrix <table><tr><td>2</td><td>-2</td><td>-4</td></tr><tr><td>-1</td><td>3</td><td>4</td></tr><tr><td>1</td><td>-2</td><td>-3</td></tr></table> is <br>(A) non-singular <br>(B) Idempotent <br>(C) Nilpotent <br>(D) Involutory
Step-by-Step Solution
Key Concept: A matrix A is idempotent if A^2 = A. Calculate A^2 by multiplying the matrix by itself and check if it equals the original matrix.
Let A = [[2, -2, -4], [-1, 3, 4], [1, -2, -3]]. <br> A^2 = [[2, -2, -4], [-1, 3, 4], [1, -2, -3]] * [[2, -2, -4], [-1, 3, 4], [1, -2, -3]] <br> = [[4+2-4, -4-6+8, -8-8+12], [-2-3+4, 2+9-8, 4+12-12], [2+2-3, -2-6+6, -4-8+9]] <br> = [[2, -2, -4], [-1, 3, 4], [1, -2, -3]] = A. <br> Since A^2 = A, the matrix is idempotent.
Correct Answer: 2