Question

Express the following matrix as the sum of a symmetric and a skew symmetric matrix

`[(4, -2),(3, -5)]`

Answer 1

For any matrix A, A + A' is always symmetric and A – A' is always skew-symmetric.

∴ A = `1/2[("A" + "A""'") + ("A" - "A""'")]`

i.e., A = `1/2("A" + "A""'") + 1/2("A" - "A""'")`

Here, A = `[(4, -2),(3, -5)]`

∴ A' = `[(4, 3),(-2, -5)]`

∴ A + A' = `[(4, -2),(3, -5)] + [(4, 3),(-2, -5)]`

= `[(8, 1),(1, -10)]`

and A – A' = `[(4, -2),(3, -5)] - [(4, 3),(-2, -5)]`

= `[(0, -5),(5, 0)]`

∴ A = `1/2[(8, 1),(1, -10)] + 1/2[(0, -5),(5, 0)]`

i.e., A = `[(4, 1/2),(1/2, -5)] + [(0, (-5)/2),(5/2, 0)]`

i.e., A is the sum of the symmetric matrix `[(4, 1/2),(1/2, -5)]` and the skew-symmetric matrix `[(0, (-5)/2),(5/2, 0)]`.