Express A is the sum of a symmetric and a skew-symmetric matrix, where [tex]A~ = \: \bigg[ \begin{matrix}3& -1 \\ 5& 2 \end{matrix} \bigg][/tex]
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Express A is the sum of a symmetric and a skew-symmetric matrix, where [tex]A~ = \: \bigg[ \begin{matrix}3& -1 \\ 5& 2 \end{matrix} \bigg][/tex]
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Answer:
Let A=[315−1], then A′=[351−1]
Now, A+A′=[315−1]+[351−1]=[666−2]
Let P=21(A+A′)=21[666−2]=[333−1]
Now, P′=[333−1]=P
Thus, P=21(A+A′) is a symmetric matrix.
Now, A−A′=[[315−1]−[351−1]=[0−440]
Let Q=21(A−A′)=21[0−440]=[0−220]
Now, Q′=[0−220]=Q
Thus, Q=21(A−A′) is a skew-symmetric matrix.
Representing A as sum of P and Q:
[315−1]−[351−1]=[0−440]
Let Q=21(A−A′)=21[0−440]=[0−220]
Now, Q′=[0−220]=Q
Thus, Q=21(A−A′) is a skew-symmetric matrix.
Representing A as sum of P and Q:
P+Q=[333−1]+[0−220]=[315−1]=A
Hope it helps
[tex]\bold{ANSWER≈}[/tex]
Let A [315-1], then A'=[351-1]
Now, A+A'=[315-1]+[351-1]-[666-2] Let P=21(A+A')=21[666-2] [333-1]
Now, P'=[333-1]=P
Thus, P=21(A+A') is a symmetric matrix. Now, A-A'-[[315-1]-[351-1]=[0-440]
Let Q=21(A-A')=21[0-440] [0-220]
Now, Q'=[0-220]=Q
Thus, Q=21(A-A') is a skew-symmetric matrix.
Representing A as sum of P and Q: [315-1]-[351-1]-[0-440]
Let Q=21(A-A')=21[0-440]-[0-220]
Now, Q'=[0-220]=Q
Thus, Q=21(A-A') is a skew-symmetric matrix.
Representing A as sum of P and Q: P+Q=[333-1]+[0-220] [315-1]=A
Hope it helps