If A and B are skew-symmetric matrices,

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Matrices

Question:

If A and B are skew-symmetric matrices, then which of the following is not true?

Options:

$A^5+ B^7$ is skew-symmetric

$A^{21}$ is skew-symmetric

$B^{18}$ is symmetric

$A^6+ B^7$ is symmetric

Correct Answer:

$A^6+ B^7$ is symmetric

Explanation:

The correct answer is Option (4) → $A^6+ B^7$ is symmetric

Given: $ A $ and $ B $ are **skew-symmetric matrices**, i.e., $ A^T = -A $ and $ B^T = -B $.

Properties used:

Odd powers of a skew-symmetric matrix are skew-symmetric: $ (A^{2n+1})^T = -A^{2n+1} $.
Even powers of a skew-symmetric matrix are symmetric: $ (A^{2n})^T = A^{2n} $.
Sum of two skew-symmetric matrices is skew-symmetric.
Sum of a symmetric and a skew-symmetric matrix is neither symmetric nor skew-symmetric in general.

Option-wise check:

$ A^5 + B^7 $: both are odd powers ⇒ both skew-symmetric ⇒ sum is skew-symmetric ⇒ True

$ A^{21} $: odd power ⇒ skew-symmetric ⇒ True

$ B^{18} $: even power ⇒ symmetric ⇒ True

$ A^6 + B^7 $: $ A^6 $ is even ⇒ symmetric, $ B^7 $ is odd ⇒ skew-symmetric ⇒ sum is neither symmetric nor skew-symmetric ⇒ False