Class 12Mathematics3 and 4 .Determinants and MatricesSpecial types of matrices, Transpose of matrices and Trace of matrices
MediumIf $A$ and $B$ are symmetric matrices,prove that $AB - BA$ is a skew-symmetric matrix.
(N/A) Given that $A$ and $B$ are symmetric matrices,we have:
$A' = A$ and $B' = B$ ........ $(1)$
We need to check the transpose of $(AB - BA)$:
$(AB - BA)' = (AB)' - (BA)'$
$= B'A' - A'B'$
$= BA - AB$ (Using equation $(1)$)
$= -(AB - BA)$
Since $(AB - BA)' = -(AB - BA)$,it follows that $(AB - BA)$ is a skew-symmetric matrix.
$A' = A$ and $B' = B$ ........ $(1)$
We need to check the transpose of $(AB - BA)$:
$(AB - BA)' = (AB)' - (BA)'$
$= B'A' - A'B'$
$= BA - AB$ (Using equation $(1)$)
$= -(AB - BA)$
Since $(AB - BA)' = -(AB - BA)$,it follows that $(AB - BA)$ is a skew-symmetric matrix.
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