Let \(M = \begin{bmatrix}1&2\\3&4\end{bmatrix}\), \(P = \begin{bmatrix}0&1\\1&0\end{bmatrix}\) and \(Q = \begin{bmatrix}1&0\\0&-1\end{bmatrix}\).
Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | \(PM\) | (I) | \(\begin{bmatrix}1&-2\\3&-4\end{bmatrix}\) |
| (B) | \(MP\) | (II) | \(\begin{bmatrix}3&4\\1&2\end{bmatrix}\) |
| (C) | \(QM\) | (III) | \(\begin{bmatrix}1&2\\-3&-4\end{bmatrix}\) |
| (D) | \(MQ\) | (IV) | \(\begin{bmatrix}2&1\\4&3\end{bmatrix}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(II), (B)-(IV), (C)-(III), (D)-(I)
Each product is found by row-by-column multiplication. Multiplying on the left changes the rows of M; multiplying on the right changes its columns.
(A) \(PM\)
This matches (II) \(\begin{bmatrix}3&4\\1&2\end{bmatrix}\).
\( \displaystyle PM = \begin{bmatrix}0 + 3&0 + 4\\1 + 0&2 + 0\end{bmatrix} = \begin{bmatrix}3&4\\1&2\end{bmatrix} \) (the rows of M are interchanged)
(B) \(MP\)
This matches (IV) \(\begin{bmatrix}2&1\\4&3\end{bmatrix}\).
\( \displaystyle MP = \begin{bmatrix}0 + 2&1 + 0\\0 + 4&3 + 0\end{bmatrix} = \begin{bmatrix}2&1\\4&3\end{bmatrix} \) (the columns of M are interchanged)
(C) \(QM\)
This matches (III) \(\begin{bmatrix}1&2\\-3&-4\end{bmatrix}\).
\( \displaystyle QM = \begin{bmatrix}1 + 0&2 + 0\\0 - 3&0 - 4\end{bmatrix} = \begin{bmatrix}1&2\\-3&-4\end{bmatrix} \) (the second row changes sign)
(D) \(MQ\)
This matches (I) \(\begin{bmatrix}1&-2\\3&-4\end{bmatrix}\).
\( \displaystyle MQ = \begin{bmatrix}1 + 0&0 - 2\\3 + 0&0 - 4\end{bmatrix} = \begin{bmatrix}1&-2\\3&-4\end{bmatrix} \) (the second column changes sign)