If $ A $ and $ B $ are skew-symmetric matrices of same order, then $ AB - BA $ is a _____

Question

For the matrices $ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} $ and $ B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} $, if $ (A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} $, then among the following which one is true?}

Answer 1

Step 1: Understanding the Concept:
A matrix $X$ is symmetric if $X^T = X$. It is skew-symmetric if $X^T = -X$. Given $A^T = -A$ and $B^T = -B$.
Step 2: Formula Application:
Let $X = AB - BA$. Take the transpose of $X$: $X^T = (AB - BA)^T = (AB)^T - (BA)^T$ Using $(PQ)^T = Q^T P^T$: $X^T = B^T A^T - A^T B^T$
Step 3: Explanation:
Substitute $A^T = -A$ and $B^T = -B$: $X^T = (-B)(-A) - (-A)(-B)$ $X^T = BA - AB = -(AB - BA) = -X$ Wait, let's re-evaluate: $X^T = (-B)(-A) - (-A)(-B) = BA - AB$. Since $X = AB - BA$, then $-X = BA - AB$. Therefore, $X^T = -X$. Correction: For skew-symmetric $A, B$, the commutator $[A, B]$ is actually skew-symmetric.
Step 4: Final Answer:
The matrix is skew-symmetric. (Option (a)).

If \( A \) and \( B \) are skew-symmetric matrices of same order, then \( AB - BA \) is a _____

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The Correct Option is A

Solution and Explanation

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