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3 and 4 .Determinants and Matrices
hard
Let $A$ and $B$ be any two $3\times3$ matrices. If $A$ is symmetric and $B$ is skewsymmetric, then the matrix $AB - BA$ is
A
skewsymmetric
B
symmetric
C
neither symmetric nor skewsymmetric
D
$I$ or $-I$, where $I$ is an identity matrix
(JEE MAIN-2014)
Solution
Let $A$ be symmetric matrix and $B$ be skew symmetric matrix.
$\therefore {A^T} = A$ and ${B^T} = – B$
Consider
${\left( {AB – BA} \right)^T} = \left( {A{B^T}} \right) – {\left( {BA} \right)^T}$
$ = {B^T}{A^T} – {A^T}{B^T}$
$ = \left( { – B} \right)\left( A \right) – \left( A \right)\left( { – B} \right)$
$ = – BA + AB = AB – BA$
This shows $AB-BA$ is symmetric matrix.
Standard 12
Mathematics
