Question

If \(A\) is any matrix and \(K\) is any constant, then \((KA)'\) is:

• A. \(K'A'\)
• B. \(A'K'\)
• C. \(KA'\)
• D. \(KA\)

Hint:

A scalar constant passes straight through a transpose: (KA)'=KA'.

Answer 1

The Correct Option is C

Step 1: Key Formula or Approach:
For a scalar \(K\) (a plain number, not a matrix) and a matrix \(A\), the transpose of a scalar multiple obeys \((KA)' = KA'\) — the scalar just carries through the transpose unchanged.

Step 2: Why it works:
Transposing only swaps rows and columns of the matrix part; multiplying every entry by the constant \(K\) does not depend on position, so the order of "multiply by K" and "transpose" can be swapped freely.

Step 3: Why the other options are wrong:
\(K'A'\) and \(A'K'\) wrongly treat \(K\) as if it were a matrix needing its own transpose; \(KA\) forgets to transpose \(A\) at all.

Final Answer:
\((KA)' = KA'\). \[ \boxed{KA'} \]