Board 2026 Class 12 Maths Q6 Solution: Which matrix is skew symmetric when A, B are

The question

If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?

(A) $AB$

(B) $AB + BA$

(C) $(A + B)^2$

(D) $A - B$

Exam answer

A and B are skew symmetric, so $A' = -A$ and $B' = -B$ ⁠.

Why?

A matrix $M$ is skew symmetric when $M' = -M$ ⁠: transposing it gives its negative.

\begin{aligned} (A - B)' &= A' - B' \\ &= -A + B \\ &= -(A - B) \end{aligned}

Why?

The transpose of a difference is the difference of the transposes. Then replace $A'$ by $-A$ and $B'$ by $-B$ ⁠, and take out the factor $-1$ ⁠.

$\therefore\ A - B$ is skew symmetric.

The other options fail in general:

\begin{aligned} (AB)' &= B'A' = BA \\ (AB + BA)' &= BA + AB \\ ((A + B)^2)' &= (A + B)^2 \end{aligned}

Why?

For a product the order reverses, and the two minus signs cancel:

(AB)' = (-B)(-A) = BA

So $AB$ would be skew symmetric only if $BA = -AB$ ⁠, which is not true in general.

For $(A + B)^2$ ⁠: $(A + B)' = -(A + B)$ ⁠, and $(-M)^2 = M^2$ ⁠. So $AB + BA$ and $(A + B)^2$ equal their own transposes: they are symmetric.

Watch out: a non-zero symmetric matrix cannot also be skew symmetric, since $M' = M$ and $M' = -M$ together give $M = O$ ⁠.

Answer: (D) $A - B$

Where this comes from

Chapter 3, Matrices: Symmetric and skew symmetric matrices.

Question type: Transpose

How to spot it: click to see

Type 8. Transpose: symmetric or skew symmetric?

Spot it: "show that $A$ is symmetric", "show that $A$ is skew symmetric", or "find $A'$ ".

Here: To test $A - B$ ⁠, write its transpose: $(A - B)' = A' - B' = -A + B = -(A - B)$ ⁠, so it is skew symmetric.

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Q6: Which matrix is skew symmetric when AAA⁠, BBB are