2 Marks Questions

Question

Express matrix as the sum of a symmetric and a skew-symmetric matrix: [ rr 1 & 5 -1 & 2 ]

Vidyadip · Accepted Answer

Given A = [ cc 1 & 5 -1 & 2 ] Explanation: As per Theorem “Any square matrix can be expressed as the sum of a symmetric and skew-symmetric matrix.” So in order to prove this we will be using Theorem which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew-symmetric matrix.” Now, Let A = [ cc 1 & 5 -1 & 2 ] Therefore, A^ = [ cc 1 & -1 5 & 2 ] Now, on adding A and A’ we will get, A+A^ = [ cc 1 & 5 -1 & 2 ]+ [ cc 1 & -1 5 & 2 ] A+A^ = [ cc 1+1 & 5+(-1) -1+5 & 2+2 ] A+A^ = [ ll 2 & 4 4 & 4 ] Now, Let M = 1 2 (A+A^ ) Therefore, M = 1 2 [ ll 2 & 4 4 & 4 ] M= [ ll 1 & 2 2 & 2 ] Now, M' = [ ll 1 & 2 2 & 2 ] M^ =M Thus M = 1 2 (A+A^ ) is a symmetric matrix as M' = M Now, on subtracting A’ from A we will get, A-A^ = [ cc 1 & 5 -1 & 2 ]- [ cc 1 & -1 5 & 2 ] A-A^ = [ cc 1-1 & 5-(-1) -1-5 & 2-2 ] A-A^ = [ cc 0 & 6 -6 & 0 ] Now, Let N = 1 2 (A-A^ ) Therefore, N = 1 2 [ cc 0 & 6 -6 & 0 ] N= [ cc 0 & 3 -3 & 0 ] Now, N' = [ cc 0 & -3 3 & 0 ] N^ =(-1) [ cc 0 & 3 -3 & 0 ] N^ =-N Thus M = 1 2 (A+A^ ) is a skew-symmetric matrix as N' = - N Now, Add M and N, we get, M+N= [ ll 1 & 2 2 & 2 ]+ [ cc 0 & 3 -3 & 0 ] M+N= [ cc 1+0 & 2+3 2+(-3) & 2+0 ] M+N= [ cc 1 & 5 -1 & 2 ] So we see here, M+N= [ cc 1 & 5 -1 & 2 ]=A Thus, A is represented as the sum of a symmetric matrix M and a skew-symmetric matrix N.

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