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

Question

Let the foot of the perpendicular from the point $ (1, 2, 4) $ on the line $ \frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3} $ be $ P $. Then the distance of $ P $ from the plane $ 3x + 4y + 12z + 23 = 0 $ is:

Answer 1

Step 1: Test on a tiny concrete matrix:
Let $A = \begin{bmatrix}1 & 2\end{bmatrix}$ (a $1\times 2$ matrix) and $K = 5$.

Step 2: Compute $(KA)'$ directly:
$KA = \begin{bmatrix}5 & 10\end{bmatrix}$, so $(KA)' = \begin{bmatrix}5\\10\end{bmatrix}$.

Step 3: Compute $KA'$ and compare:
$A' = \begin{bmatrix}1\\2\end{bmatrix}$, so $KA' = 5\begin{bmatrix}1\\2\end{bmatrix} = \begin{bmatrix}5\\10\end{bmatrix}$, which matches $(KA)'$ exactly.

Final Answer:
The check confirms $(KA)' = KA'$. \[ \boxed{KA'} \]

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

Show Hint

The Correct Option is C

Solution and Explanation

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