Step 1 (Governing Laws of Matrix Transpose): [0.25 Mark]

According to the properties of the transpose of a matrix as defined in the NCERT textbook, for any matrices $A$ and $B$ of suitable orders and a scalar $k$, the following fundamental laws hold:

• Property of Addition: $(A + B)' = A' + B'$
• Property of Subtraction: $(A - B)' = A' - B'$
• Property of Multiplication (Reversal Law): $(AB)' = B'A'$
• Property of Scalar Multiplication: $(kA)' = kA'$

Step 2 (Step-by-Step Mathematical Verification): [0.5 Mark]

We evaluate each given statement against the established laws:

• (i) $(A + B)' = A' + B'$: This is a standard property of transpose. (True)
• (ii) $(A - B)' = B' - A': Since $(A - B)' = A' - B'$, and $A' - B' \neq B' - A'$ (matrix subtraction is not commutative), this statement is (False).
• (iii) $(AB)' = A'B': According to the Reversal Law of transpose, $(AB)' = B'A'$. Therefore, $A'B'$ is generally incorrect. (False).
• (iv) $(kAB)' = kB'A': Using the properties $(kA)' = kA'$ and $(AB)' = B'A'$, we get: $$(kAB)' = k(AB)' = k(B'A') = kB'A'$$ This statement is (True).

Step 3 (Final Concluded Answer): [0.25 Mark]

Based on the verification above, only statements (i) and (iv) are true.

Correct Option: (D)

💡 EL10 Concept Insight: Think of the transpose like flipping a pancake; when you flip a stack of two pancakes ($AB$), you have to flip the top one and the bottom one, but they also swap their positions, which is why $(AB)'$ becomes $B'A'$ instead of $A'B'$.