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

Step 1: Check Statement I
A matrix $A$ is non-singular if: \[ |A| \neq 0 \] Inverse of a matrix is: \[ A^{-1} = \frac{1}{|A|}\,\text{adj}(A) \] Since $|A| \neq 0$, $A^{-1}$ exists.
\[ \Rightarrow \text{Statement I is true} \]
Step 2: Check Statement II
Given $A$ and $B$ are symmetric: \[ A' = A, B' = B \]
Step 3: Take transpose of $(AB - BA)$
\[ (AB - BA)' = (AB)' - (BA)' \] \[ = B'A' - A'B' \] \[ = BA - AB \]
Step 4: Compare
\[ (AB - BA)' = -(AB - BA) \]
Step 5: Conclusion
Since $M' = -M$, matrix is skew-symmetric.
\[ \Rightarrow \text{Statement II is true} \] Final Answer:
\[ \boxed{\text{Option (3)}} \]