Question:
If A be a square matrix, such that \((A – A^T)^T = A^T – (A^T) = A^T – A\)
यदि A एक वर्ग आव्यूह (square matrix) है, ताकि \((A – A^T)^T = A^T – (A^T) = A^T – A\) हो, तो
Options:
- \(A – A^T\) is not a square matrix
- \(A – A^T\) is a square matrix
- \(A – A^T\) is a symmetric matrix
- \(A – A^T\) is a skew symmetric matrix
- \(A – A^T\) एक वर्ग आव्यूह नहीं है
- \(A – A^T\) एक वर्ग आव्यूह है
- \(A – A^T\) एक सममित आव्यूह है
- \(A – A^T\) एक विषम-सममित आव्यूह है
Correct Answer: d)
Explanation:
Let \(B = A – A^T\). Taking transpose: \(B^T = (A – A^T)^T = A^T – (A^T)^T = A^T – A = -(A – A^T) = -B\). Since \(B^T = -B\), \(A – A^T\) is a skew-symmetric matrix.
माना \(B = A – A^T\)। परिवर्त (transpose) लेने पर: \(B^T = (A – A^T)^T = A^T – (A^T)^T = A^T – A = -(A – A^T) = -B\)। चूँकि \(B^T = -B\) है, अतः \(A – A^T\) एक विषम-सममित आव्यूह (skew-symmetric matrix) है।