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The inverse of a skew-symmetric matrix (if it exists) is:

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A symmetric matrix

A skew symmetric matrix

A Diagonal matrix

None of matrix is unique

answer is B.

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Detailed Solution

We have,

A ′ = − A ................. (1)

We know that,

A A − 1 = A − 1 A = I .............. (2)

Where

I

is the identity matrix and

A − 1

is the inverse of

A

Now taking transpose on each side of equation (2) we get,

A A − 1 ′ = A − 1 A ′ = (I) ′

Since, transpose of Identity matrix is Identity matrix so,

A A − 1 ′ = A − 1 A ′ = I

Applying Reversal Law of transpose of matrices i.e.

(A B) ′ = B ′ A ′

we get,

A − 1 ′ A ′ = A ′ A − 1 ′ = I

A − 1 ′ (− A) = (− A) A − 1 ′ = I

(Since from equation (1),

A ′ = − A

)
From the above equation we see that

A − 1 ′ (− A) = I

which means that

A − 1 ′

is the inverse of

(− A)

(Since, their product is identity matrix which implies that one is inverse of the other)

⇒ A − 1 ′ = − A − 1

Now from the above equation, we see that

A − 1

satisfies equation 1 i.e. the condition of a skew symmetric matrix.
Hence,

A − 1

is skew symmetric i.e. the inverse of a skew symmetric matrix is a skew symmetric matrix.

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