Question

Consider the following statements:
Statement I : If $A$ is a non-singular matrix, then $A^{-1}$ exists.
Statement II : If $A$ and $B$ are symmetric matrices of same order, then $(AB - BA)$ is a skew symmetric matrix.
Choose the correct option.

• A. Statement I is true and Statement II is false
• B. Statement I is false and Statement II is false
• C. Statement I is true and Statement II is true
• D. Statement I is false and Statement II is true

Hint:

The matrix expression $(AB - BA)$ is a very important construct in linear algebra called the commutator. The property proved here (that the commutator of two symmetric matrices is skew-symmetric) is a common standard result worth remembering.

Answer 1

The Correct Option is C

Let's analyze each statement given in the problem:

1. Statement I: If \(A\) is a non-singular matrix, then \(A^{-1}\) exists.
   • A matrix is termed "non-singular" if its determinant is non-zero, i.e., \(\det(A) \neq 0\). For a non-singular matrix, there always exists an inverse, denoted as \(A^{-1}\).
   • Therefore, Statement I is true.
2. Statement II: If \(A\) and \(B\) are symmetric matrices of the same order, then \((AB - BA)\) is a skew-symmetric matrix.
   • A matrix \(C\) is skew-symmetric if \(C^T = -C\).
   • For symmetric matrices \(A\) and \(B\), we have \(A^T = A\) and \(B^T = B\).
   • Consider \((AB - BA)^T\):
     \((AB - BA)^T = (AB)^T - (BA)^T = B^T A^T - A^T B^T = BA - AB\).
   • We find that \((AB - BA)^T = -(AB - BA)\). Hence, \((AB - BA)\) is a skew-symmetric matrix according to the definition.
   • Therefore, Statement II is true.

Both statements are true based on the above reasoning.

Conclusion: The correct option is: \(Statement\ I\ is\ true\ and\ Statement\ II\ is\ true\).