Question

21
If B is a symmetric matrix, check whether the matrix ABA^Tis symmetric
or skew symmetric.
​

Answer 1

Basic Definition :-

Symmetric Matrix :-

A square matrix A is said to be symmetric iff

$\boxed{ \red{ \bf \: {A}^{T} = A}}$

Skew - Symmetric Matrix :-

A square matrix A is said to be skew - symmetric iff

$\boxed{ \red{ \bf \: {A}^{T} = - A}}$

Solution :-

Given that,

$\rm :\longmapsto\:B \: is \: symmetric \: matrix$

$\bf\implies \: {B}^{T} = B - - - (1)$

Now,

We have to check that,

$\rm :\longmapsto\:AB {A}^{T}\: is \: symmetric \: or \: skew - symmetric.$

So,

Consider,

$\rm :\longmapsto\: {( AB{A}^{T}) }^{T}$

$\sf \: = \: \: {( {A}^{T})}^{T} {B}^{T} {A}^{T}$

$\sf \: = \: \: AB{A}^{T}$

$\bf\implies \:\: {(AB {A}^{T}) }^{T} = AB{A}^{T}$

$\bf\implies \:\: AB{A}^{T} \: is \: symmetric.$

Additional Information :-

$\boxed{ \red{ \bf \: {(A + B)}^{T} = {A}^{T} + {B}^{T}}}$

$\boxed{ \red{ \bf \: {(A - B)}^{T} = {A}^{T} - {B}^{T}}}$

$\boxed{ \red{ \bf \: {(AB)}^{T} = {B}^{T} {A}^{T}}}$

$\boxed{ \red{ \bf \: {( {A}^{T})}^{T} = A}}$

$\boxed{ \red{ \bf \: {(kA)}^{T} = k \: {A}^{T}}}$