Question

Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix `[(4, -2),(3, -5)]`.

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

∴ A^T = `[(4, 3),(-2, -5)]`

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

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

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

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

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

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

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

= `(1)/(2)[(8, 1),(1, -10)]`

= `[(4, 1/2),(1/2, -5)]`
and
Q = `(1)/(2)("A" - "A"^"T")`

= `(1)/(2)[(0, -5),(5, 0)]`

= `[(0, -(5)/(2)),(5/2, 0)]`

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

and Q is a skew-symmetric matrix.   ...[∵ a_ij = – a_ij]
∴ A = P + Q

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