Let \(A = \begin{bmatrix}1&-1\\2&3\end{bmatrix}\).
Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | The sum of the diagonal elements of \(A^2\) | (I) | 6 |
| (B) | The element in the first row and second column of \(A + A'\) | (II) | 10 |
| (C) | The element in the second row and second column of \(A'A\) | (III) | 1 |
| (D) | The value of \(k\) for which \(A - A' = k\begin{bmatrix}0&1\\-1&0\end{bmatrix}\) | (IV) | \(-3\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(I), (B)-(III), (C)-(II), (D)-(IV)
The transpose of A is
\( \displaystyle A' = \begin{bmatrix}1&2\\-1&3\end{bmatrix} \)
and each product is worked out row by column.
(A) The sum of the diagonal elements of \(A^2\)
This matches (I) 6.
\( \displaystyle A^2 = \begin{bmatrix}1 - 2&-1 - 3\\2 + 6&-2 + 9\end{bmatrix} = \begin{bmatrix}-1&-4\\8&7\end{bmatrix} \)
\(-1 + 7 = 6\)
(B) The element in the first row and second column of \(A + A'\)
This matches (III) 1.
\(a_{12} + a_{21} = -1 + 2 = 1\)
(C) The element in the second row and second column of \(A'A\)
This matches (II) 10. The second row of \(A'\) is (−1, 3) and the second column of A is (−1, 3).
\((-1)(-1) + 3(3) = 1 + 9 = 10\)
(D) The value of \(k\) for which \(A - A' = k\begin{bmatrix}0&1\\-1&0\end{bmatrix}\)
This matches (IV) \(-3\).
\( \displaystyle A - A' = \begin{bmatrix}0&-3\\3&0\end{bmatrix} = -3\begin{bmatrix}0&1\\-1&0\end{bmatrix} \)