Josalynn

2021-02-08

Let A be a $nxxn$ matrix and let

$B=A+{A}^{T}andC=A-{A}^{T}$

(a) Show that B is symmetric and C is skew symmetric.

(b) Show that every n × n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.

(a) Show that B is symmetric and C is skew symmetric.

(b) Show that every n × n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.

Nicole Conner

Skilled2021-02-09Added 97 answers

as we know that a matrix E is said to be symmetric if ${E}^{T}=E$ and skew symmetric if ${E}^{T}=-E.$

as $B=A+{A}^{T}$ therefore,

$={B}^{T}=(A+{A}^{T}{)}^{T}$

$={A}^{T}+({A}^{T}{)}^{T}$

$={A}^{T}+A(As({A}^{T}{)}^{T}=A)$

=B

as ${B}^{T}=B$ therefore, B is symmetric matrix.

hence proved.

As C=A−AT

therefore, $CT=(A-{A}^{T}{)}^{T}$

$={A}^{T}-({A}^{T})T$

$={A}^{T}-A$

$=-(A-{A}^{T})$

=−C

as ${C}^{T}=-C$ therefore, C is skew symmetric matrix. Hence proved.

Now we have to show that every nxxn matrix can be expressed as the sum of the symmetric and skew symmetric matrix. Let A be nxxn matrix. therefore,

$A=1/2(2A)$

$=1/2(A+A)$

$=1/2(A+{A}^{T}+A-{A}^{T})$

$=1/2(B+C)$

where $B=A+{A}^{T}$ and B is a symmetric matrix and $C=A-{A}^{T}$ and C is a skew symmetric matrix.

herefore,

$A=1/2(B+C)$

$=1/2$(symmetric matrix +skew symmetric matrix)

therefore, it has been showed that any matrix A of order nxxn can be expressed as the sum of symmetric and skew symmetric matrix.

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