First slide

Transpose of matrix

Question

Which of the following is correct?

Moderate

A

If A is a symmetric matrix, then Aⁿ is symmetric, n ∈ N
B

If A is a skew-symmetric matrix then Aⁿ is symmetric if n is even, n ∈ N
C

If A is a skew-symmetric matrix then Aⁿ is skew-symmetric if n is odd, n ∈ N
D

All of these

Solution

Let A be a symmetric matrix.
Then, (Aⁿ)′ = (AAA … n times)′
= (A′ A′ A′ … A′ n times)
= (A′ )ⁿ = Aⁿ ( $∵$ A′ = A)
$∴$ Aⁿ is also a symmetric matrix.
Now, let A be a skew-symmetric matrix. Then A′ = –A.
We have, (Aⁿ)′ = (AAA … n times)
= (A′ A′ A′ … A′ n times) [ $∵$ (AB)′ = B′ A′]
= (A′ )ⁿ = (– A)ⁿ ( $∵$ A′ = –A)
= (–1)ⁿ Aⁿ
$= (\begin{matrix} A^{n} i f n i s e v e n \\ - A^{n} i f n i s o d d \end{matrix})$
Hence, Aⁿ is symmetric if n is even and skew-symmetric if n is odd.
The correct option is (A), (B), (C) and (D)

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