If mathrm{A}, ,mathrm{B} are symmetric matrices of same order, then mathrm{A B}-mathrm{B A} is a

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3 and 4 .Determinants and Matrices

easy

If $\mathrm{A}, \,\mathrm{B}$ are symmetric matrices of same order, then $\mathrm{A B}-\mathrm{B A}$ is a

A

Zero matrix

B

Symmetric matrix

C

Skew symmetric matrix

D

Identity matrix

Solution

$A$ and $B$ are symmetric matrices, therefore, we have :

$A^{\prime}=A$ and $B^{\prime}=B$ ………. $(1)$

Consider $(A B-B A)^{\prime} =(A B)^{\prime}-(B A)^{\prime}$ $[\because $ $=A^{\prime} -B^{\prime}] $

$=B^{\prime} A^{\prime}-A^{\prime} B^{\prime}$ $ [ \because $ $B^{\prime} A^{\prime}]$

$=B A-A B $ $[$ by $(1)$ $]$

$=-\,(A B-B A)$

$\therefore $ $(A B-A B)^{\prime} =-(A B-B A)$

Thus, $(A B-B A)$ is a skew-symmetric matrix.

Standard 12

Mathematics

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