# Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix [4-23-5]. - Mathematics and Statistics

Sum

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

#### Solution

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)]

∴ AT = [(4, 3),(-2, -5)]

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

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

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

Also, A – AT = [(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         ...[∵ aij = aij]

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

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

Concept: Algebra of Matrices
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