Match each matrix P in List-I with its square \(P^2\) in List-II, where I is the identity matrix and O is the zero matrix of order 2.
| List-I (P) | List-II (\(P^2\)) | ||
|---|---|---|---|
| (A) | \(\begin{bmatrix}1&0\\2&-1\end{bmatrix}\) | (I) | O |
| (B) | \(\begin{bmatrix}2&-2\\1&-1\end{bmatrix}\) | (II) | I |
| (C) | \(\begin{bmatrix}2&4\\-1&-2\end{bmatrix}\) | (III) | \(-I\) |
| (D) | \(\begin{bmatrix}1&2\\-1&-1\end{bmatrix}\) | (IV) | P |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
Each square is found by row-by-column multiplication of P with itself.
(A) \(\begin{bmatrix}1&0\\2&-1\end{bmatrix}\)
This matches (II) I.
\( \displaystyle P^2 = \begin{bmatrix}1 + 0&0 + 0\\2 - 2&0 + 1\end{bmatrix} = \begin{bmatrix}1&0\\0&1\end{bmatrix} \)
(B) \(\begin{bmatrix}2&-2\\1&-1\end{bmatrix}\)
This matches (IV) P.
\( \displaystyle P^2 = \begin{bmatrix}4 - 2&-4 + 2\\2 - 1&-2 + 1\end{bmatrix} = \begin{bmatrix}2&-2\\1&-1\end{bmatrix} \)
(C) \(\begin{bmatrix}2&4\\-1&-2\end{bmatrix}\)
This matches (I) O.
\( \displaystyle P^2 = \begin{bmatrix}4 - 4&8 - 8\\-2 + 2&-4 + 4\end{bmatrix} = \begin{bmatrix}0&0\\0&0\end{bmatrix} \)
(D) \(\begin{bmatrix}1&2\\-1&-1\end{bmatrix}\)
This matches (III) \(-I\).
\( \displaystyle P^2 = \begin{bmatrix}1 - 2&2 - 2\\-1 + 1&-2 + 1\end{bmatrix} = \begin{bmatrix}-1&0\\0&-1\end{bmatrix} \)