Matrices and Determinants
Symmetric and Skew-Symmetric Matrices
IIT-JEE 2002
Grade 12
Question:
If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that $A + B$ is defined, then $AB - BA$ is always:
(1) a symmetric matrix
(2) a skew-symmetric matrix
(3) a diagonal matrix
(4) the zero matrix
Step-by-Step Solution
Key Concept: Take the transpose of $AB - BA$ termwise, using $(XY)^T = Y^T X^T$ together with $A^T = A$ and $B^T = -B$, and see exactly what expression you are left with.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)