If A is a square matrix then A - A T is a skew hermitian matrix-A- TrueB- False

The correct option is A True Lets take general element from matrix A and verify if A A^T is a skew symmetric matrix. Assume aij = p + iq aji = r + is The expre ...

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Standard IX

Mathematics

Skew Hermitian Matrix

If A is a squ...

Question

If A is a square matrix then $A -^{T}$ is a skew hermitian matrix.

True

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False

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Solution

The correct option is A
True

Lets take general element from matrix A and verify if $A -^{T}$ is a skew symmetric matrix.

Assume

aij = p + iq

aji = r + is

The expression $A -^{T}$ will translate to aij $= - ¯ ¯¯¯¯¯ ¯ a j i$ at the elementary level in the resultant matrix.

If mij is the ijth element in resultant matrix

mij = $a i j - ¯ ¯¯¯¯¯ ¯ a i j = (p + i q) - (r - i s)$

=(p – r)+ i (q + s) ….(1)

mji = $a i j - ¯ ¯¯¯¯¯ ¯ a i j = (r + i s) - (p - i q)$

= r – p + I (q+s) ….(2)

From (1) & (2) we can see that

$m i j = - ¯ ¯¯¯¯¯¯¯ ¯ m j i$

Thisimplies that the resultant matrix or $A -^{T}$ is skew hermitian.

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If A is a square matrix then A−¯¯¯¯¯¯¯AT is a skew hermitian matrix.

If A is a square matrix then $A -^{T}$ is a skew hermitian matrix.