Let \(P = \begin{bmatrix}0&1\\1&0\end{bmatrix}\) and \(Q = \begin{bmatrix}1&0\\0&-1\end{bmatrix}\), and let I and O be the identity and zero matrices of order 2.
Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | \(PQ\) | (I) | \(2I\) |
| (B) | \(QP\) | (II) | \(O\) |
| (C) | \(PQ + QP\) | (III) | \(\begin{bmatrix}0&1\\-1&0\end{bmatrix}\) |
| (D) | \(P^2 + Q^2\) | (IV) | \(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
Each product is worked out row by column; PQ and QP turn out different, so the order of multiplication matters.
(A) \(PQ\)
This matches (IV) \(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\).
\( \displaystyle PQ = \begin{bmatrix}0(1) + 1(0)&0(0) + 1(-1)\\1(1) + 0(0)&1(0) + 0(-1)\end{bmatrix} = \begin{bmatrix}0&-1\\1&0\end{bmatrix} \)
(B) \(QP\)
This matches (III) \(\begin{bmatrix}0&1\\-1&0\end{bmatrix}\).
\( \displaystyle QP = \begin{bmatrix}1(0) + 0(1)&1(1) + 0(0)\\0(0) + (-1)(1)&0(1) + (-1)(0)\end{bmatrix} = \begin{bmatrix}0&1\\-1&0\end{bmatrix} \)
(C) \(PQ + QP\)
This matches (II) O.
\( \displaystyle PQ + QP = \begin{bmatrix}0 + 0&-1 + 1\\1 - 1&0 + 0\end{bmatrix} = O \)
(D) \(P^2 + Q^2\)
This matches (I) 2I.
\( \displaystyle P^2 = \begin{bmatrix}0 + 1&0 + 0\\0 + 0&1 + 0\end{bmatrix} = I \)
\( \displaystyle Q^2 = \begin{bmatrix}1 + 0&0 + 0\\0 + 0&0 + 1\end{bmatrix} = I \)
\(P^2 + Q^2 = I + I = 2I\)