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Question
If `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q is equal to ______.
Options
`[(2, 5//2),(5//2, 4)]`
`[(0, 5//2),(-5//2, 0)]`
`[(0, -5//2),(5//2, 0)]`
`[(2, -5//2),(5//2, 4)]`
Solution
If `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q is equal to `underlinebb([(0, -5//2),(5//2, 0)])`.
Explanation:
Given `[(2, 0),(5, 4)]` = P + Q
For any matrix A, we have
A = `1/2 [(A + A^') + (A - A^')]`
= `(A + A^')/2 + (A - A^')/2`
where, `(A - A^')/2` is a symmetric matrix i.e., Q,
∴ Q = `1/2{[(2, 0),(5, 4)]-[(2, 5),(0, 4)]}`
= `1/2[(0, -5),(5, 0)]`
= `[(0, -5//2),(5//2, 0)]`
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