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Express the matrix A=[4 2-1 3 5 7 1-2 1]...

Express the matrix `A=[4 2-1 3 5 7 1-2 1]` as the sum of a symmetric and a skew-symmetric matrix.

Text Solution

Verified by Experts

A matrix `A` can be expressed as the sum of symmetric and skew-symmetric matrix in following way: `A = 1/2(A+A^T)+1/2(A-A^T)`
Here,
`A = [[4,2,-1],[3,5,7],[1,-2,1]]`
`:. A^T = [[4,3,1],[2,5,-2],[-1,7,1]]`
`=>A+A^T = [[4,2,-1],[3,5,7],[1,-2,1]]+ [[4,3,1],[2,5,-2],[-1,7,1]] = [[8,5,0],[5,10,5],[0,5,2]]`
`=>A-A^T = [[4,2,-1],[3,5,7],[1,-2,1]]- [[4,3,1],[2,5,-2],[-1,7,1]] = [[0,-1,-2],[1,0,9],[2,-9,0]]`
`:. A = 1/2[[8,5,0],[5,10,5],[0,5,2]]+1/2[[0,-1,-2],[1,0,9],[2,-9,0]]`
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