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COMEDK 2025 Afternoon Solved Paper

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Question : 140 of 180

Marks: +1, -0

If

A=\left[\begin{array}{ccc}0 & -1 & 2 \\ 1 & 0 & 3 \\ -2 & -3 & 0\end{array}\right]

A+2 A^T=

$-A^T$
$A^T$
$2 A^2$
$A$

Solution:

Concept:

Recognize that

A

is a skew-symmetric matrix, meaning

A^T = -A

.

Explanation:

First, compute the transpose of

A

:

A = \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & 3 \\ -2 & -3 & 0 \end{pmatrix} \implies A^T = \begin{pmatrix} 0 & 1 & -2 \\ -1 & 0 & -3 \\ 2 & 3 & 0 \end{pmatrix}.

Note that

A^T = -A

, since every element of

A^T

is the negative of the corresponding element in

A

.

Now evaluate

A + 2 A^T

:

Substitute

A^T = -A

into the expression:

A + 2 A^T = A + 2(-A) = A - 2A = -A.

But

-A = A^T

from the skew-symmetric property.

Thus

A + 2 A^T = A^T

.

Answer:

Option B:

A^T

.

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