If A is a square matrix then,
is symmetric matrix
is skew-symmetric matrix
All of the above
Properties of symmetric and skew symmetric matrices -
Properties of symmetric and skew-symmetric matrices:
i) if A is a square matrix, then AA’ and A’A are symmetric matrices
ii) All positive integral power of symmetric matrices are symmetric matrices, because
iii) If A is a symmetric matrix, then -A, kA, A’, An, A-1, B’AB are also symmetric matrix where n ∈ N, k ∈ R and is B a square matrix of order same as matrix A.
iv) If A is a skew-symmetric matrix then
A2n is a symmetric matrix for n ? N.
A2n+1 is a skew-symmetric matrix for n ? N
kA is also a skew-symmetric matrix, where k ∈ R
B’AB is also a skew-symmetric matrix where B a square matrix of order same as matrix A
v) If A and B are symmetric matrices then:
A ± B, AB+BA are symmetric matrices.
AB - BA is a skew-symmetric matrix.
AB is a symmetric matrix iff AB = BA.
iv) If A and B are skew-symmetric matrices then:
A ± B, AB - BA are skew-symmetric matrices.
AB + BA is a symmetric matrix.
vi) If A is a skew-symmetric matrix and C is a column matrix, then C’AC is a zero matrix.
-
Let
now,
which is a symmetric matrix
, which is a skew-symmetric matrix
Hence all the options are correct
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