Question

If A and B are skew symmetric matrices, prove that AB + BA is symmetric matrix.

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

A is skew symmetric matrix.

$\rm\implies \:A' \: = \: -\: A$

B is skew symmetric matrix.

$\rm\implies \:B' \: = \: - \: B$

Now, we have to prove that AB + BA is symmetric matrix.

It means we have to prove that (AB + BA)' = AB + BA

Now, Consider

$\rm \: (AB + BA)'$

$\rm \: = \: (AB)' + (BA)'$

$\rm \: = \: B'A' + A'B'$

$\rm \: = \: ( - B)( - A) + ( - A)( - B)$

$\rm \: = \: BA + AB$

$\rm \: = \: AB + BA$

$\rm\implies \:(AB + BA)' = AB + BA$

$\rm\implies \:AB + BA \: is \: symmetric \: matrix.$

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Basic Concept Used

Skew - Symmetric matrix :- A square matrix is said to be skew - symmetric iff A ' = - A

Symmetric matrix :- A square matrix A is said to be symmetric iff A ' = A

(A + B)' = A' + B'

(AB)' = B'A'

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ADDITIONAL INFORMATION

Properties of Transpose

$\rm \: (A - B)' = A' - B'$

$\rm \: (A')' = A$

$\rm \: (kA)' = kA' \: \: where \: k \: is \: non \: zero \: real \: number$

Answer 2

Solution -

Given that,

A is skew symmetric matrix

» A' = - A

B is skew symmetric matrix

» B' = - B

Now, we have to prove that AB + BA is rectify symmetrix metrix.

It means we have to prove that

(AB + BA)' = AB + BA

Now, Consider

(AB + BA)'

= (AB)' + (BA)'

= B'A' + A'B'

= (-B)(-A) + (-A) (-B)

= BA + AB

= AB + BA

» (AB + BA)' = AB + BA''

» AB + BA is symmetrix metrix.

Hence proved