Question

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

`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`

Answer 1

A square matrix A can be expressed as the sum of a symmetric and a skew-symmetric matrix as

A = `1/2("A" + "A"^"T") + 1/2("A" - "A"^"T")`

Let A = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`

∴ A^T = `[(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`

∴ A + A^T = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)] + [(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`

= `[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`

Also, A – A^T = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)] - [(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`

= `[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`

Let P = `1/2("A" + "A"^"T") = [(3, 1/2, (-5)/2),(1/2, -2, -2),((-5)/2, -2, 2)]`

and Q = `1/2("A" - "A"^"T") = [(0, 5/2, 3/2),((-5)/2, 0, 3),((-3)/2, -3, 0)]`

∴ P is a symmetric matrix  ...[∵ a_ij = a_ij]

and Q is a skew symmetric matrix 

∴ A = P + Q  ...[∵ a_ij = – a_ij]

∴ A = `[(3, 1/2, (-5)/2),(1/2, -2, -2),((-5)/2, -2, 2)] + [(0, 5/2, 3/2),((-5)/2, 0, 3),((-3)/2, -3, 0)]`