Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade Class 11
Question:
Let $A = [a_{ij}]_{n \times n}$, $n$ is odd natural number. Then determinant of matrix $(A - A^T)^{2015}$ is _____.
Step-by-Step Solution
Key Concept: Any skew-symmetric matrix of odd order must have determinant zero because $|A| = (-1)^n|A|$ for odd $n$ forces $|A| = 0$.
For a skew-symmetric matrix $A$ of odd order $n$, we have $A^T = -A$, so $|A^T| = |A|$. Also $|A^T| = (-1)^n|A| = -|A|$ for odd $n$. Therefore $|A| = -|A|$, which implies $|A| = 0$. Thus $(A - A^T)^{2015} = 0$.
<div class="key-concept"><strong>Key Concept:</strong> Any skew-symmetric matrix of odd order must have determinant zero because $|A| = (-1)^n|A|$ for odd $n$ forces $|A| = 0$.</div>
<div class="trap-box"><strong>Trap:</strong> Students may forget that the determinant property $|A^T| = (-1)^n|A|$ applies specifically to skew-symmetric matrices only when $n$ is odd.</div>
Correct Answer: 0